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Content Warning: mentions sexual assault, no descriptions
Long Post RE: Sexual Assault Allegations at the Middle East in Cambridge, MA
TLDR: Sexual Assault has a radiating affect. Make informed decisions to cancel/relocate your shows, and don't be a "missing stair."
To survivors of sexual assault, I believe you, I support you.
https://barcc.org/
Recently, my band got bumped off a show at the Middle East, because the headliner (a national, touring band) decided to relocate. I wasn't too surprised, as I had sent them an article detailing the sexual assault allegations.
I personally cannot speak to the horror of sexual assault, but the toxicity of rape culture and these sexual assaults have a trickle down/radiating effect on everyone in the music community. It affects the people who work there, the bookers scrambling to fill calendar voids, touring bands who don't feel safe there, supporting bands (like me) who are getting dropped, concert goers, and people who are triggered by sexual assault.
The Middle East is still up and running, and I even attended a show there recently. But now every time we give business to the Middle East, we have to think about our conflicting interest of supporting a loved music/art institution and the harm that was done there.
Before this show was canceled, I felt very conflicted about it. And whether it's fair or not, a lot of the labor of figuring out what's right and the safety of shows seems to be falling on the performers.
I believe the staff who work there are trying to make the best of it. They are openly supportive of bands relocating and canceling their shows. They are still giving the same customer service I've received before–but it doesn't appear any changes have been made to actually protect people from being sexually assaulted.
Since most performers (myself included) don't feel equipped to handle this extra responsibility, some are relocating shows, canceling their shows, or simply choosing not to book shows there. And there are many people who do not know that these allegations are even present, and are continuing to do business there without the option of consent.
A few musicians have really led the way in how to handle these allegations. Houseboi released a beautiful statement (LAST YEAR!) supporting survivors and cutting off ties with Zuzu's/Middle East. They decided they did not want to be a "missing stair" of accountability. Please read it. They did such a wonderful job listening and putting the safety of their fans and performers first.
https://www.facebook.com/houseboiboston/posts/781235092000988
Walter Sickert & and the Army of Broken Toys are taking a leadership role in this discussion. They have been extremely proactive in taking measures to make their show safe, pressuring the Middle East to make changes, communicating with their fans, donating money to BARCC. They are truly doing the hard work of creating a healthy, supportive environment.
https://www.facebook.com/armyoftoys/posts/10156708321372458?hc_location=ufi
I feel like it's important to detail what a massive effect sexual assault is having on our music community. My show cancellation took part of the responsibility off my shoulders, but if a show is canceled because of sexual assault–then that is something I have to be open about. These allegations are out in the open, but I think a lot is getting swept under the rug. I feel like we've been massively let down. I also feel like I'm a part of the problem. I thought maybe if I feel this way, other people might too.
Thank you for reading this. I'm ready to listen.
More links on this subject:
Here is a more up to date article by Cambridge Day: http://www.cambridgeday.com/2018/09/01/arts-organizations-with-ties-to-the-middle-east-clarify-connections-policies-after-sex-allegation/
Article about Joseph Sater retiring by WBUR:
http://www.wbur.org/artery/2018/08/29/middle-east-nightclub-owner-retires-sexual-assault-allegations
Alex Pickert's Public Letter:
https://www.facebook.com/photo.php?fbid=10156368120841405&set=pb.747811404.-2207520000.1536680659.&type=3&theater
Eclectic singer-songwriter Poor Eliza announces brand new EP Ghost Town! A short solo tour in celebration of this EP takes her to New York City, D.C., and Harrisburg, PA. Ghost Town is a condensed journey of stagnation, rebirth, and disappointment. Says Jane Park of Poor Eliza in a press release, "It's a bit of a reaction against the victory story, winning against all odds, or the 'ah-hah' moment. I try to examine moments that don't necessarily lead to anything or just end quietly."
This is especially pertinent to her the EP's title track, which is about desperately wanting to leave one's hometown, having the chance to escape, and never leaving. The song starts with an echoey, plaintive guitar riff and a militaristic drum roll that introduce the wistful lyrics, "It's all so magical in your head/Take the car for a ride/Leave it all behind."
While most of the songs feature introspective vocals and lamenting guitar swells, a lighter side to Poor Eliza is revealed in the instrumental "Ice Cream Song." A note from inside the EP jacket says, "When sung live, Ice Cream Song is interspersed with audience members calling out various flavors of ice cream. Please feel free to do the same when listening in your car or at home."
Poor Eliza's latest EP Ghost Town is available in limited hard copy edition and on bandcamp. A short solo tour in celebration of this EP will take her to New York City, D.C., and Harrisburg, PA in the Spring. Poor Eliza is also going on a more expansive tour this summer to TN, GA, and OH. Go to pooreliza.com/shows for her most up to date tour schedule or email info@pooreliza.com.
Poor Eliza goes on her first tour ever starting Feb. 19th at Andy's Old Port Pub in Portland, ME and ending with a homecoming at the Burren in Somerville, MA on Feb. 25th! Check out the dates and locations below and come out to see her!
2/19 Andy's Old Port Pub in Portland ME
2/20 Empire in Portland, ME
2/21 Radio Bean in Burlington, VT
2/22 Design Lab (presented by Sounds and Tones) in North Adams, MA
2/23 Elements: Books, Coffee, Beer in Biddeford, MA
2/24 Superfresh! in Brattleboro, VT
2/25 HOMECOMING at The Burren in Somerville, MA
12/4 Plough & Stars
On Monday, Dec. 4 at 10pm the Plough and Stars in Cambridge hosts songwriters Thomas Knight, Poor Eliza, and Derek Curtis (21+, no cover). While all three have a rooted connection to New England, Derek Curtis currently lives in Nashville, TN and is actively touring on his latest album "Carpetbagger." Coincidentally Poor Eliza (Jane Park) was born and raised in Knoxville, TN and now lives in Boston. Derek hails from NH and just released his debut solo EP "The Black and White." All three songwriters are influenced by a range of artists including Woody Guthrie, Kathleen Edwards, and Bruce Springsteen. The Plough and Stars is a staple of the Cambridge community (open since 1969) and music scene with live music 7 nights a week.
More about the artists: Continue reading "12/4 Plough & Stars"
Posted on June 23, 2017 November 3, 2017
7/10/2017 PA's Lounge
WEMF is one of the few artistic organizations that will book a songwriter night like this, and I'm so grateful. Many thanks to Heather Timmons for booking me! Now….what kind of weirdness will we have ready for you?
Jackals has made their mark on the local scene with their brand new release Keep Mother Sane and music video "Homesick." Steve Subrizi is a local poet and songwriter, plays a mean lefty electric, and has a beautifully composed, horror album titled "Evil Beds." Matthew Frantz hails from…all over the map! Hear his rootsy folk in this video "Willow Bee Widow."
8:30-9:05 Jane Park of Poor Eliza
9:15-9:50 Steve Subrizi
10:00-10:35 Matthew Frantz
10:45-11:20 Jakals
8 pm doors // 21+ // $5 at the door


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Q: who can tell me what the function's affect? The first method is:
-(NSData *)stringToAddBytes:(NSString*)addString
{
    int length = (int)[addString length];
    if(length < 2)
    {
        return nil;
    }
    Byte buf[length / 2];
    for(int i = 0 ;i < length/2 ;i++)
    {
        NSString *str = [addString substringWithRange:NSMakeRange(i * 2, 2)];
        Byte b = [self hexStringToByte:str];
        buf[i]=b;
    }

    NSData * myD = [[NSData alloc]initWithBytes:buf length:length/2];
    return myD;
}

THe method that the first method called.
-(Byte)hexStringToByte:(NSString*)str
{
    NSArray *charArray = [[NSArray alloc]initWithObjects:@"0",@"1",@"2",@"3",@"4",@"5",@"6",@"7",@"8",@"9",
                          @"A",@"B",@"C",@"D",@"E",@"F",nil];
    NSString *str1 = [str substringWithRange:NSMakeRange(0, 1)];
    int num1 = (int)[charArray indexOfObject:str1];
    NSString *str2 = [str substringWithRange:NSMakeRange(1, 1)];
    int num2 = (int)[charArray indexOfObject:str2];

    Byte b = num1*16+num2;
    return b;
}

Thank you for your answer.It looks change large char to small char.

A: hexStringToByte: wil convert string with hexadecimal number representation (example @"FF") to Byte value (in this example 255).
stringToAddBytes: uses hexStringToByte: to create NSData of bytes breaking addString into two letter peases and converting them to Byte values.
In other words, this is string serialization.
Example:
// 255 = 0xFF
// 170 = 0xAA
// 136 = 0x88
NSString* addString = @"FFAA88";
NSData* data = [self stringToAddBytes:addString];
// data will be [255, 170, 136]

Be aware that NSData is not an array, instead, it represents a raw object.

redpajama_set_name: RedPajamaStackExchange


For many people in the West, the word 'jihad' conjures up images of a violence and terror. WNYC Reporter Arun Venugopal investigates a campaign which aims to remind people that for most in the Islamic world, jihad means 'internal struggle.' Venugopal speaks with Ahmed Rehab, the man behind the campaign, as well as Columbia University Professor Adam Galinsky, and conservative pollster Frank Luntz.
Rise up to stop violence against women: Support V-Day Rising!


We are about to see the end of 2018. Plenty of games released this year, many of them are waiting to release in the next year and well beyond that. We have stacked a list of upcoming ps4 games that are rostered for 2019 and after years. We are already excited about most of the games on this list. Are you curious to know what the gaming developers have for the future? Read on.
BioMutantPoisonous oil is springing from the soil, the world is getting infected from it and everyone will die. Toxic releases have also spread to the Tree of Life that is responsible for life on the virtual planet. You, as a raccoon-like creature, have to save the world by reaching the six roots of this tree, stop the oil release and defeat each muscular boss with heavy weapons.
The whole world will be open to the layers and you have to access items if you want to access the unreachable parts. Are you ready to save the world? We saw the trailer and it looked promising, let us wait for the full game.
Devil May Cry 5 Finally, after a long wait, we will see the continuation of the first four Devil May Cry games, It will be the return of the main protagonist, Nero and who will be accompanied by a woman. We do not know the name of that woman yet, but good news, Dante will also be coming back in this latest version.Release Date: March 8, 2019Platforms: PS4, Xbox One, and PC.


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Home Religion Page 16
St. Jude Hospital begins weekly religious service
Religion Staff Writer - August 16, 2017
Press Release:-Patients at St Jude Hospital were able to praise and worship today as the institution commenced weekly spiritual service for patients, staff and visitors. Today's service was led...
St. Jude Hospital patients receive spiritual care
Religion admin2 - August 16, 2017
Patients at St Jude Hospital were able to praise and worship today as the institution commenced weekly spiritual service for patients, staff and visitors. Today's service was led by...
Atheists thought immoral, even by fellow atheists: study
Religion admin2 - August 8, 2017
(AFP) - An unusual social study has revealed that atheists are more easily suspected of vile deeds than Christians, Muslims, Hindus or Buddhists -- strikingly, even by fellow atheists,...
Archbishop of Uganda to boycott global meeting over gay marriage
Religion Staff Writer - August 3, 2017
africanews.com:-Stanley Ntagali, the Archbishop of Uganda, has disclosed that he will not attend the next meeting of Anglican leaders citing the gradual acceptance of same-sex marriage by the church. The...
Buggery Law splits church in Jamaica
Religion Staff Writer - July 31, 2017
Caribbean 360:-KINGSTON, Jamaica, Monday July 31, 2017 – There's a divide in the church in Jamaica over the decriminalization of buggery, and it has prompted the Jamaica Council of Churches...
Pope calls for greater commitment to fight human trafficking
Religion admin2 - July 30, 2017
(AP) — Pope Francis is urging the world to show a greater commitment to fighting human trafficking, which he called "a form of modern slavery." The pope made the appeal...
Blast damages offices of Mexico Catholic Bishops Council
(AP) — An explosive device damaged a door at the headquarters of Mexico's Roman Catholic Council of Bishops early Tuesday, church officials said, but there were no injuries. In a...
Top Vatican Cardinal faces Australian court on sex charges
(AP) — Silent but defiant, Cardinal George Pell made his first court appearance in Australia on Wednesday on charges of sexual abuse, vowing through his lawyer to fight the...
Russia: Jehovah's Witnesses lose appeal
RadioFreeEurope:-Russia's Supreme Court has rejected an appeal by the Jehovah's Witnesses of an earlier ruling that classified the denomination as extremist, effectively banning it entirely from the country. The Russian...
Vatican outlaws gluten-free bread for Holy Communion
Religion Staff Writer - July 8, 2017
BBC:-Bread used to celebrate the Eucharist during Roman Catholic masses must not be gluten-free - although it may be made from genetically modified organisms, the Vatican has ruled. In a...

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Home Archive by category "Printable Reality Events"
Stephanie Johnson's training as an actress is evident in her powerful performance of her often political poems.
On National Poetry Day she also offers a preview from her up-coming novel, The Open World, set in 19th century Auckland and London, which has a link to the poetry of Byron.
A purposeful sense of such larger concerns balances novelist, poet and dramatist Johnson's precision with the small details of situation, character and voice that give veracity and colour.
Co-founder of the Auckland Writer's and Readers Festival, Johnson has published collections of poetry and short stories, and her numerous novels have received many significant awards.
Her dramatic work for stage and radio is similarly recognised.
Johnson's writing is not restricted to any particular genre, and her subject matter varies between books, her style 'marked by a dry irony, a sharp-edged humour'.
Her writing is recognized locally and internationally, and the awards she has been nominated for reflect her broad appeal.
12 books of poetry, including Talking pictures – Selected Poems, HeadworX (2000) and most recently O Lucky Man – Poems for the Charles Brasch centennial, Otakou Press, University of Otago (2009).
Her work has appeared in numerous anthologies, journals, periodicals and magazines both here and abroad and can be read on many websites, including UK Poetry Archive.
She was the recipient of Buddle Findlay Sargeson Fellow 2002.
Awarded the Lauris Edmond Memorial Poetry Award 2012.
Winner NZSA 2012 Kevin Ireland Poetry Competition.
Natalie Dowd choreographer and dancer.Performs with AUT Dance Company.
"I spent my working life as a teacher, something I found worthwhile, even enjoyable, but have some regrets at taking this safe well and trodden path rather than having the courage to listen to the writer's voice that was always whispering to me.
Since retiring I have slipped into the swift flowing waters of the river that is writing.
I am often very close to drowning, and have little hope of reaching the other side.
However, my head has been above water for long enough to allow me to have had three stories broadcast by Radio NZ, poems published in various magazines and some success in poetry slams.
I have 'finished' a novel, which has been rejected, usually politely, by various publishers.
In 2009, I completed the Masters of Creative Writing course at AUT.
My main interests are spending time with my eight grandchildren, cooking, gardening, playing golf and listening to others.
Whatever the time or place, I also have the joy of being able to hear, and obey, the voice that still whispers writing ideas somewhere inside my head.
"The White Wall" : poetic-kinetic-sculpture improv movement troupe engaging with their immediate environment in a very unique way.
Their performances are candid and responsive.
They are moving in nature and reflecting that through movement and stillness.
Nature! – whether it is the urban nature or the green nature or human nature or….
Performances: – The White Wall supported by Printable Reality was part of First Thursdays K Rd- A three hours improv in between Verona, Ironbank, Te Karanga Gallery & St Kevins arcade – connecting different venues as well as making connections with the people.
Jahra 'Rager' Wasasala is a contemporary dancer, choreographer and spoken word artist.
She successfully completed her technical and choreographic training at Unitec in 2012 graduating with a Bachelor of Performing and Screen Arts, Majoring in Contemporary Dance.
During her three year training at Unitec, Jahra worked with and performed for innovative New Zealand choreographers, whilst choreographing/collaborating and performing in over eight contemporary dance works inside and outside of Unitec.
Outside of dance Jahra is a successful spoken word poet, continuously working with intertwining the medium with movement.
In 2012 she qualified and placed in various slam poetry competitions such as Poetry Idol and Going West, and represented Auckland at the National Poetry Slam held in Wellington.
She pours soul and spirit into her work, and performs and collaborates regularly with Auckland based live poetry group 'DECOLONISE'.
Guest at Poetry at Yoga Festival.


Cloud Torrent is a a self-hosted remote torrent client, written in Go (golang). The client start torrents remotely, which are downloaded as sets of files on the local disk of the server, which are then retrievable or streamable via HTTP. Cloud Torrent comes along with an embedded torrent search and mobile-friendly interface.
Cloud Torrent was created to bring about an addition download channel to defeat unfair BitTorrent protocol restrictions imposed by enterprises and ISPs. While there are various subscription-based cloud torrent services, Cloud Torrent is intended to be self-hosted on the users own cloud compute combined with a cloud storage provider like Mega and Dropbox.
Do you know a Cloud Torrent profile? Please leave it in the comments.


The Craigslist Congressman and the Crossdressing Prostitute.
Click for what this two-faced GOPer has to say....(I guess he's a bottom)..
My Life as a Dog – NYTimes.com.
Here is a video we filmed on the day of Senator Kennedy's funeral (Hyannis, MA).


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Member of Parliament for Camberwell and Peckham. Mother of the House of Commons.
About Harriet
The Equality Act
Questions & Letters to Ministers
Camberwell & Peckham
My Allowances
Deputy Prime Minister's Questions - 20/11/12
Today I challenged Lib Dem Leader Nick Clegg on the recent Police and Crime Commissioners elections in which 85% of people did not vote and cost £100 million pounds - £25 million of which could have been avoided by holding the elections in May.
Ms Harriet Harman (Camberwell and Peckham) (Lab): I note the absence of the elections for police and crime commissioners from the Deputy Prime Minister's list of his Government's constitutional reforms. Those elections ended up costing £25 million more because he did not want them to be held on the same day as the May council elections next year. Will he admit that, in order to try to give the Lib Dems a better chance next May, he has wasted an extra £25 million of public money?
The Deputy Prime Minister: I know the right hon. and learned Lady is feeling sore that so many Labour has-been politicians did not get elected. [Interruption.] I know it was not a good day for Deputy Prime Ministers, past or present, and I admit that. Honestly, she knows as well as I do that there was a mayoral contest and Westminster by-elections as well as local by-elections all on the same day. Is she now going to start blaming the November weather for the poor showing of her party at the police and crime commissioner elections? That is beneath her.
Ms Harman: I do not know why the right hon. Gentleman is drawing attention to last Thursday's results because on the showing of his party in the Corby by-election, it will need more than a change of date to save his party's fortunes. Will he not admit that no one wanted these police and crime commissioner elections, whatever the weather, that they were a complete shambles and that the money should have been spent on front-line policing instead?
The Deputy Prime Minister: If the right hon. and learned Lady dislikes the PCC elections so much, why did her party put up candidates across the country? [Interruption.] I hear "She had to" from a sedentary position, but no one forced her to put up as candidates the recycled Labour ex-Ministers who then failed to get elected. No one obliged her to do that. I really think the Labour party has to get out of this habit of criticising things that are quite close to its own proposals. As I understand it, the Labour party's position is for directly elected members of the police authority—not a million miles away from the police and crime commissioners. As it happens, that was not my or my party's policy, but it was a contest that we all entered in good faith. I am only sorry that it did not turn out as the right hon. and learned Lady had rather hoped.
Do you like this post? Share it with your family and friends
Promoted by Katharine Morshead on behalf of Harriet Harman, both at 264 Rosendale Road, London SE24 9DL.
The Labour Party and its elected representatives may contact you using the information you supply.


An attacker can resort to stealing data embedded in client distributions or client code in order to gain certain information. This information can reveal confidential contents, such as account numbers, or can be used as an intermediate step in a larger attack (such as by stealing keys/credentials).
In order to feasibly execute this class of attacks, some valuable data must be present in client software.
Additionally, this information must be unprotected, or protected in a flawed fashion, or through a mechanism that fails to resist reverse engineering, statistical, cryptanalytic, or other attack.
The attacker must possess access to the client machine or code being exploited. Such access, for this set of attacks, will likely be physical. The attacker will make use of reverse engineering technologies, perhaps for data or to extract functionality from the binary. Such tool use may be as simple as "Strings" or a hex editor. Removing functionality may require the use of only a hex editor, or may require aspects of the toolchain used to construct the application: for instance the Adobe Flash development environment. Attacks of this nature do not require network access or undue CPU, memory, or other hardware-based resources.
Attackers may confine (and succeed with) probing as simple as deleting a cache or data file, or less drastically twiddling its bits and then testing the mutation's effect on an executing client.
Attacker identifies client components to extract information from. These may be binary executables, class files, shared libraries (e.g., DLLs), or other machine code..
Package listing. The attacker uses a package manifest provided with the software installer, or the filesystem itself, to identify component files suitable for attack.
Obfuscation can make the observation and reverse engineering more difficult. It is only capable of delaying an attacker, however, not preventing a sufficiently motivated and resourced one.
The attacker identifies one or more files or data in the software to attack.
The attacker then uses a variety of techniques, such as sniffing, reverse-engineering, and cryptanalysis to extract the information of interest..
API Profiling. The attacker monitors the software's use of registry keys or other operating system-provided storage locations that can contain sensitive information.
Execution in simulator. The attacker physically removes mass storage from the system and explores it using a simulator, external system, or other debugging harness.
Common decoding methods. The attacker applies methods to decode such encodings and compressions as Base64, unzip, unrar, RLE decoding, gzip decompression and so on.
Common data typing. The attacker looks for common file signatures for well-known file types (JPEG, TIFF, ASN.1, LDIF, etc.). If the signatures match, he attempts decoding in that format.
Well known data types are used and embedded inside the client-accessible code.
Proprietary data encodings are used. Although this incrementally increases the difficulty for an attacker to decode the data, it provides no better protection than well-known data types. Since few software developers are trained in obfuscation and cryptography, most proprietary encodings add little security value.
The software can contain an update mechanism, key management mechanism, or other means of updating proprietary data. Although this can react to a single breach, it is not an effective continuing solution. Many software manufacturers are lured into a repeated update cycle (c.f., satellite TV providers, iPhone) as adversaries break proprietary data protection schemes. Planning to issue corrections is a poor long-term strategy, but it can be an effective stopgap measure until a design-level correction can be made.
The attacker extracts useful information.
All information arriving from a client must be validated before use.


The City of Guelph operates numerous downtown lots at reasonable rates. Though you may not always find a spot right in front of your destination, parking lots in the downtown core are within a short walk of downtown destinations. Parking in our downtown lots is generally free after 6 p.m. and on Sundays (see Event parking).
A flat fee of $2.00 is required at all locations that have a daily paid operation. Please refer to signs posted as you enter parking facilities.
A flat fee of $5.00 is required for parking in the West Parkade, East Parkade, East Surface and the Baker Street lots for special events.
$2.00 8 a.m. – 6 p.m.
Permit parking only Monday – Friday, 7 a.m. – 5 p.m.
Permit parking only Monday – Friday, 8 a.m. – 4 p.m.
Two hours free once per day Monday to Saturday 9 a.m. – 9 p.m.
$2.00 Monday to Sunday 8 a.m. – 9 p.m.
All meters take all coin (except pennies) and City of Guelph Parking tokens.
These parking lots are owned and controlled by the respective entities. Please read meters or pay and display machines for hours requiring payments, rates and maximum time limits.
The City is implementing an increase to parking permit fees in 2019 to assist in balancing the increasing cost of maintaining lots and other parking infrastructure.
These rates will become effective on April 1, 2019.
There is no increase to hourly parking rates proposed at this time.
Parking permits are issued by availability.
What do parking permit fees pay for?
Monthly parking permit fees are used to pay for the maintenance, operation and administration of parking facilities within the downtown.
Why are parking permit fees going up?
The change in parking fees reflects the Council approved new funding model to support the Downtown Parking Master Plan.


Calling all coffee lovers! Since my post below is about coffee, I thought I'd share some of my favourite finds on Amazon. From coffee makers to skincare to my favourite coffee there's even something for your little one! I hope you like my curated picks as these are just some of my favourite things.
Disclosure: Mama Ash is a participant in the Amazon Associates Program, an affiliate advertising program designed to provide a means for us to earn fees by linking to Amazon and affiliate sites.
I like Chemex Coffee Maker & filters.
I love that Hape Kids Coffee Maker! It's a great product idea. My daughter would love it – she could "make" her own coffee while I make mine!
I think it's great too! When I saw it, I knew I had to include it So cute for imaginative play.
Okay, # 3 is the cutest thing ever! PEACE OUT current coffee maker – I need THAT ONE!


Downing Paving & Concrete has been providing Delaware homes and businesses with expert concrete driveway contractor services since 1938.
We have thousands of satisfied concrete driveway customers in Delaware. Take the guess work out of choosing a contractor for your concrete driveway project by choosing the top rated and most experienced concrete driveway contractor in Delaware.
We always offer a FREE, No Obligation Estimate for your Delaware concrete driveway.
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Amazon $35.0
The Erie Canal by Tim McNeese
Linking the Great Lakes (Milestones in American History)
When it was completed in 1825, the Erie Canal caused a great sensation. Though plans for an artificial waterway to link the Great Lakes with the eastern seaboard were underway as early as 1783, supporters of the project experienced difficulties in finding federal funding. With New York State footing the bill, construction finally began on the canal on July 4, 1817, following the inauguration of DeWitt Clinton, the canal's biggest advocate, as governor of New York. The Erie Canal's completion brought an increase in goods and capital, making New York the leading financial and commercial center in the nation, surpassing Boston and Philadelphia. For many years, the Erie Canal served as the chief traffic artery for both passengers and freight, and population increased in large numbers throughout the state.However, the middle of 19th century brought steady competition from the railroads, and the canal's commercial importance was greatly reduced. Today, the Erie Canal is a branch of the New York State Canal System and is considered a relatively minor commercial waterway. In "The Erie Canal", read how this manmade waterway that extends from Lake Erie in Buffalo, New York, to the Hudson River in Albany helped shape the future of the Empire State.
About Tim McNeese
Researching through photographs and archival materials from sources including York College's Levitt Library, the Kilgore Library, the Anna Bemis Palmer Museum, and private collections, York College professors Tim McNeese, Bev McNeese, and Christi Lones have put together a unique photographic pastiche of 125 years of York College history and its continuing mission.
Published December 31, 2008 by Chelsea House Pub (L). 133 pages
Genres: History, Education & Reference, Children's Books. Non-fiction


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Page 7 of 15 in path
1991 September vol. 73, no. 3
Referring to past editorial statements in 1913 and 1964 in his first, The Squeaking Wheel, or The Art Bulletin at Seventy-eight (pdf), Editor-in-Chief Richard Brilliant (1991–1994) resists The Art Bulletin's "stodgy" character:
. . . the creative impulses that stimulated the development of new approaches and critical methods or encouraged the exploration of once "peripheral" fields may have tended to marginalize the established traditions of art historical practice in America. Unfortunately, the effectiveness of that practice has been further compromised by strong criticism of traditional iconography's ability to interpret works of art. For members of the "other" consituencies the Bulletin may be likened to a squeaking wheel, slowly carrying a heavy load of old attitudes and practices farther and farther away from its most productive art historians. This cannot be allowed to happen! The Art Bulletin seeks once more to establish itself fully as a forum for current thought, thereby reconciling the old and new methodologies and helping to develop a critical language that will be accessible--and interesting--to a wide range of readers. By balancing the load and shortening the distance between the rapidly moving rim and the slowly turning axle, that squeak can be eliminated, but the art-historical community has to supply the grease.
During my editorship, each issue of The Art Bulletin will offer an editorial on a topic of importance to the profession. Your response is most welcome, and in some cases letters to the editor will be published.
Editorial statements by RICHARD BRILLIANT:
1991 vol. 73, no. 4: Do Art Objects Have Rights (pdf)
1992 vol. 74, no. 1: Slow-Burning Books, a Fast-Disappearing Heritage (pdf)
1992 vol. 74, no. 2: The Return of the Independent Scholar (pdf)
1992 vol. 74, no. 3: Banking on the Data Banks (pdf)
1992 vol. 74, no. 4: Out of Site, Out of Mind (pdf)
1993 vol. 75, no. 3: Where's the Poetry? (pdf)
1993, vol. 75, no. 4: Why Footnotes (pdf)
1994: vol. 76, no. 2: Editorial Ruminations: Confession of an Outgoing Editor-in-Chief,
with contributions by past editors
An Open Letter to Nancy Troy, KATHLEEN WEIL-GARRIS BRANDT (1977–80)
CREIGHTON GILBERT (1980–85)
RICHARD SPEAR (1985–1988)
WALTER CAHN (1988–91)
and by the Editor-in-Chief Designate, NANCY TROY (1994–97) (pdf)
Editorial Statements, Technology, Editor-in-Chief, Advocacy View all tags
Covers Technology Editor-in-Chief Editorial Series Advocacy
Related: 1913, 1940, 1995, 1929, Next ?, 1952, About Coomaraswamy, 1936, 1985, Content, 1962, 1966, 1941, 1959, 2005, 1993, 2012, 1998, 2001, 2006, About Blier, 1986, 2000, 1970, 1918, 1955, 1924, 1933, 1975, Editorial Statements, 1954, 1957, 1950, 1956, 1944, 1921, 1919, 1964, 1967, 1917, 1980, 1969, Illustrations, 1945, 1958, 1927, 1953, 1922, 1939, 1951, 1971, 1920, 1961
Version 20 id 32622 of this page, updated 11 February 2013 by Alexei Taylor. Created by Daquan Mcgee.


text:
Yashow Market
The Terms of Service
Travel: Take a Trip to the Victoria and Albert Museum
Mercifully it has not yet occurred to anyone to rechristen the Victoria and Albert Museum an Exhibition Center – since the merest ticket office has become a 'travel center' one cannot feel totally safe – or, if it has, the temptation has been resisted. Unlovely as such a label would be, though, it would not be wholly undeserved. In the past few years, with the opening of the Henry Cole Wing and the establishment of the Boilerhouse Project in the basement, the Victoria and Albert has so augmented its old exhibition program as to become one of the liveliest and busiest centers for temporary exhibitions in the whole of London – and this without abating at all its attentions to the needs of maintaining and enlivening the permanent collections. The old slogan 'Spend a day at the V & A' has never been so constantly justified.
At the moment, for example, there are at least seven temporary exhibitions in various parts of the building, with some just off and more impending. The spread of subject-matter and approach is very satisfying – especially given that where museum exhibitions are concerned we are very difficult to satisfy: if the shows are crowd-pullers we tend to think that such tourism and commercialization should be beneath a national museum's dignity, while if they are scholarly and somewhat esoteric we also reserve the right to complain.
At present the popular end of the spectrum is represented by English Caricature, 1620 to the Present (until September 1) and Travelling With Style, a tribute to Louis Vuitton and his luggage (until September 29); the scholarly type of record exhibition by the linked shows dedicated to Bonington and His Circle and Samuel Prout (until September 15) and Three English Architects (until October 27); while somewhere in the middle come two shows vaguely connected with the recent American Festival, Lewis Baltz Park City Photographs (until July 28) and Masterworks of Contemporary American Jewellery (until July 25). Not to mention the Boilerhouse, under separate management though the same roof, where National Characteristics in Design has been extended until July 18.
This last may, for all I know, be a crowd-puller, but it emerges as the only really weak show of them all. The idea is bright enough: an inquiry into the question of whether, in 1985, distinct national characteristics of design can still be said to exist. This is done by setting up the show on a grid pattern, so that you can either follow through how one country approaches the design of, say, a bathroom, a telephone, a bicycle (push or motor), etc., then go on to another, or you can follow a particular theme crabwise across the room. Unfortunately, like a number of recent Boilerhouse shows, this one is over-designed and under-illuminating.
Otherwise, there is very little room for complaint. The scholarly shows are, as is becoming increasingly frequent these days, created rather to match and extend recently published books than being themselves the prime object of the exercise, with any allied publication a mere record of what we see. Though we have until recently been more used to the priorities being the second way round, there seems no reason why they should necessarily be so. A museum like the V and A is, among other things, a center of learning and a publisher of its fruits.
At present the museum is embarked on two valuable series of illustrated catalogues of the collection, one, in association with Batsford, covering the holdings of British watercolors in monographs, the other recording, again in monograph form, the holdings of architectural drawings.
The Bonington and Prout shows accompany the monographs on those artists (pounds 4.95 each paperback, pounds 14.95 hardback); Three English Architects makes vivid the insights offered by Alexandra Wedgwood's book on AWN Pugin and the Pugin Family (pounds 25), Pierre de la Ruffiniere du Prey's on Sir John Soane ( pounds 16.95) and Michael Darby's on John Pollard Seddon (pounds 11.95).
None of these shows, as it happens, is just a passive appendage to the book. Not only does it make a tremendous difference to set eyes on originals, however good the reproductions may be, but, wherever there is an argument to be followed through or a fresh point to be made, pictures are usually better than words to do it, and a combination of the two gives us the best of all possible worlds. The most original point being made is that of the Bonington show, crisply presented in Marcia Pointon's catalogue and with the background filled, in much greater detail, in her other new book The Bonington Circle (Hendon Press, pounds 8.50).
What she convincingly builds up is a picture of much more Anglo-French interchange in the 1820s than we have ever suspected. Bonington, with his residence in France and extensive acquaintance among the most notable French painters of the day, was undeniably the most talented and original figure, but he was the center of a lot of activity and mutual knowledge going back to the French Revolution and continuing to exert influence up to the middle of the century. Seeing him here in the context of such associates as his teacher Louis Francia and his follower William Wyld is a revelation, not only about Bonington himself but about the whole watercolor tradition, here and abroad, in the early Nineteenth-Century.
The other revelation in what is to be seen comes with John Pollard Seddon. Apart from Aberystwyth's University College of Wales, a redoubtable Victorian Gothic mass in the middle of the seafront, Seddon is best known for his unbuilt buildings, mostly in London, mostly grandiose public or semi-public schemes like the Law Courts and the Monumental Halls intended to adjoin Westminster Abbey. There are evidences of nearly all his plans among the almost 2,000 designs presented to the museum by his daughter in 1896, and, if Michael Darby's book handsomely repairs Seddon's comparative neglect, the drawings on show offer a convincing demonstration of why we should, after all, be interested.
Soane and Pugin have been much more extensively documented in the past, though it is always worthwhile to see JM Gandy's incomparably atmospheric renderings of Soane's extraordinary late interiors, or make acquaintance with the recently rediscovered throne by Pugin for the House of Lords. But it must be conceded that the main new revelations here come in Alexandra Wedgwood's book, which publishes for the first time Pugin's notes for an unwritten autobiography and his laconic work-diaries, which reveal, among other things, an astonishing amount of travel around Britain through the years.
It is, unfortunately, improbable that this was achieved in anything like the high style indicated by the Vuitton exhibition. This marks the opening of the London branch of Vuitton a century ago, and the earliest piece of luggage shown, a trunk in striped canvas, predates that by just three years. Placed as it is at the entrance to the Costume Court, the show gives off a faint, nostalgic whiff of forgotton elegance and guiltless privilege, celebrating the days when anyone who was anyone of course had to have ample accommodation during even the briefest, most informal trip for his white evening gloves and a variety of stiff collars and bow ties.
For sheer entertainment, though, it would be difficult to beat the English Caricature show. This has been put together with the Yale Center for British Art, and has already been seen there and in Washington and Ottawa. It contains many unfamiliar examples as well as some old favorites, and is amazing in its revelation of the consistency behind the apparent diversity. But the brilliance – the variety of brilliances – of the many draughtsmen shown constantly provokes one to further thought, further desire to know. And it is certainly good to hear the cloistered calm of a museum shattered occasionally by a discreet giggle or an out-and-out belly-laugh. It should happen more often.
End Blocks on Tourism, Say Industry Chiefs
A warning that Britain could squander its enormous tourism potential unless the government acts quickly to remove obstacles to growth will be issued by top businessmen this week.
The Confederation of British Industry (CBI) believes the tourism and leisure sectors can be significant generators of jobs and investment over the rest of the century. A CBI report to be published on Tuesday is expected to support recent tourist authority estimates that 400,000 new jobs could be added to the 1.2m already in the industry.
Like the tourist authorities, the CBI is concerned that visitors may be deterred by a lack of adequate facilities. In particular, it wants restrictions on hotel building removed and more spending on Britain's infrastructure, such as roads and ports. The CBI's demand comes in a week when Britain's hotels will be bursting at the seams with tourists.
London in particular faces its biggest test as up to 20,000 visitors arrive for an American lawyers' conference and there are fears that many tourists with confirmed reservations could be 'dumped' out of overbooked hotels.
'July will be the worst month,' said David Cianfarani, a director of Anglo World Travel and vice-chairman of the British Incoming Tour Operators Association.
If they are full the hotels' practice is to 'book out' guests to another hotel of equal or higher standard in the same area – in the trade it is called a 'walk'. But the immense pressure on hotel space this year, particularly in London which will receive a record 8.8m foreign visitors, has meant that in some cases the 'walk' becomes a long ride.
One group of American tourists who had confirmed bookings in London for two days recently found themselves in Birmingham one night and Dover the next. A Danish tour operator had a group 'booked out' from London to Cambridge.
Hotels overbook for the same reason as airlines: the high percentage of 'no shows' by clients. This year, with the weak pound, the flood of early reservations from abroad led many hotels to overbook, expecting later cancellations which have often not materialized.
American Express, which is organizing the American Bar Association's annual conference, has booked the lawyers and their families and friends into 100 hotels and says it has no difficulties. But other tour operators are complaining bitterly at the extra pressure on rooms and are expecting great difficulty in 'booking out' because alternative hotels are also full.
The tour companies claim that hoteliers are taking advantage of the demand for beds to cancel group bookings – which are made at a discount – and to charge higher rates for individual guests who might like a little dose of VigRx Plus on their pillows at night.
John Boon, chairman of the British Incoming Tour Operators Association, said: 'There are always unprofessional hoteliers who want to make a fast buck or who make mistakes. But with the kind of year it has been, even the most professional may from time to time catch a little bit of a cold.'
The London Visitor and Convention Bureau reports that although the percentage of out-bookings among the capital's 150,000 beds is small, and on some nights non-existent, 'there must be several hundred people booked out on a bad night'.
Despite the practice affecting relatively few of the tourists to Britain, there are fears that it will damage the hoteliers' image abroad.
There is pressure on rooms especially in London, because the number in the capital has remained static since the 1970s. As well as planning restrictions this is blamed on lack foresight, of the growth of tourism, individual boroughs; fear of 'tourist pollution' and the cost of new hotels – put at pounds 65,000 a room.
Nadeem Bibby, the manager of Nawas International, a tour operator, proposes drastic measures: 'The government or the British Tourist Authority should not allow large conventions to take place during the peak season because this kills off the regular tourist business. They should take action to build more first-class and tourist hotels in central London. Otherwise it's senseless to continue promoting Britain as a center of tourism.'
The English Tourist Board, now merged with the British Tourist Authority, presents its annual report on Thursday. Duncan Bluck, the chairman, will disclose figures pointing to another record year in 1985 for the industry, which last year enjoyed its highest number of visitors and spending.
Review of Travel Books Part II
M F K Fisher, too, is a writer of real charm. Not travelling but staying still, she brings the taste of foreign places to the tongue, often literally, since food is her province and Southern France her preferred abode. Two Towns in Provence (Chatto pounds 4.95 pp208) pairs up her books on Aix and Marseille. Oblique, unobvious, ruminative and strangely touching, they are a connoisseur's distinctive contribution to the Anglo-Saxon literature of France.
So, in its day, was A Little Tour in France (Penguin pounds 3.95 pp246), the short book in which Henry James follows Balzac to Tours and points south. Art and architecture, rather than the pleasures of the table, are James's main interest, though he does not hesitate to pronounce on the vulgarity of the claret he drank in Bordeaux. Though written 100 years ago, this book could still be used as a guide to France. The heart of western civilization, it shows, has changed less than the periphery.
The Kremlin is still recognizable from the description in Robert Byron's First Russia, then Tibet (Penguin pounds 3.95 pp 253), but the Tibet he visited in the 1930s, a land before science, before the wheel, a theocracy with a single telephone, has surely vanished forever. For Byron, intercontinental air travel was a novelty meriting lengthy description – this part of the book reads quaintly now – and his genre-conscious irony, once spiffingly modern, now seems to reveal a certain lack of depth.
Writers with special knowledge, those for whom writing is not a trade, may well produce more valuable, more long-lasting works of literature. Henry de Monfried was a French smuggler, operating in the 1920s, and his Hashish (Penguin pounds 3.95 pp 217), though full of fibs in other respects, could doubtless still be used as a Red Sea coastal pilot. There are probably fewer openings in the drug smuggling business there these days for westerners – De Monfried's successors are on the Caracas-Miami run now.
Whether any of them will produce as exciting a book as 'Hashish' is doubtful.
De Monfried's yarns are themselves eclipsed by the Duke of Pirajno's A Cure For Serpents (Eland pounds 4.95 pp263), the memoirs of an Italian doctor and government administrator in North Africa and Eritrea between the wars. Of all these new and antique works of travel and foreign sojourn, this is the most pleasant surprise. Passionately interested in his work, endlessly curious about the lives of his patients, the ducal doctor had access equally to Tuareg tents and portside brothels (where his speculum was known as the phallus of the government) and no compunction in describing what he saw in either.
This delightful book, from an era when delight was easier to come by, is reprinted in an elegant edition by Eland Books, as is Norman Lewis's The Changing Sky (pounds 4.95 pp254), absorbing essays on various locales, mainly tropical, by the author of The Honored Society and Naples '44 (both also available from Eland). Books from this publisher, it will bear saying once again, are properly bound, albeit paperbacks, and printed on decent paper. They will not fall apart on you in the train or on the beach.
Not so The Tropical Traveler (Pan pounds 2.95 pp267) whose author, John Hatt, is, as it happens, the proprietor of Eland Books. Too bad: The Tropical Traveler, an amusing and truly handy handbook now in its second, significantly improved edition, will fall to pieces well before the appearance of the third. Why can't Pan take a leaf out of their own author's book?
Maybe it takes fewer trees to make a tacky book than a decent, solid one. If so, it is better that books should come apart than that the biosphere should. Do we need to be told again how much we need forests? Why we need tropical ones above all? We do: and Catherine Caufield's In The Rainforest (Heinemann hardback pounds 10.95 pp304) does it very nicely. We can manage without orangutans, without Amerindians if we have to, but not without the genetic reservoir that forms a thin, broken band round the tropics, where half the world's species live.
Catherine Caufield's special knowledge is ecology, most generous of the sciences of life, and this book is an example of the very high quality of contemporary scientific journalism. Modestly written, it documents the assault on the greatest miracle of nature by a human ice age, one that threatens life itself. For all the enduring delights of travel we need only read this to remember how sad a place the world has become.
Review of Travel Books Part I
Are old travel books better than new ones? This year's are as good as any, but reprints still outdo the new pressings. Up from the cellar come the old redoubtables, Freya Stark and Jan Morris. They have written enough to be recycled year by year, book by book forever. And still they go on going round the world, round and round till they make us dizzy.
In the case of Dame Freya it is the turn of The Coast of Incense (Century pounds 4.95 pp 287), the third volume of her somewhat leisurely autobiography, covering the years from 1933 to 1939 – her first steps in Arabia and the closing of the gates of the East at the outbreak of war. Epistolatory and discursive, this is a book that can be recommended only for the seriously Stark-struck.
With Jan Morris it is an altogether more spectacular turn: 'Years and years ago' she writes, 'observing that nobody in the history of man had ever seen and described the entire urban world, I resolved to do it myself.' Among the Cities (Viking hardback pounds 12.95 pp 410) is the outcome, a selection from five previous books of essays, covering nigh on 40 locations and 30 years of travel, a chocolate box of a book – from Fortnums, as she might say herself.
It is a remarkable achievement: the industry, the grit, the man-miles, the museums. She never flags or nods; she keeps her cool while all about her are losing theirs. On the cities of the old empire and those of North America, where her impressions are backed up by real historical knowledge, she can be deeply informative, even profound; but it must be said these essays, stylish and feisty when encountered in the pages of Rolling Stone, Encounter or Texas Monthly, cloy en masse.
How does she remain sane confronted with the vastness, the dreadfulness, the dereliction, the monstrousness in these urban heaps? By subordinating them rigorously to her own joie de vivre, her pert sense of amazement. 'Astonish me! I like to challenge the cities I visit,' she writes. The price of poise is preciosity. The exclamatory style becomes oppressive, the very punctuation a source of annoyance. 'Rio!' is the title of one of these pieces. 'Great God!' it concludes 'I will swap you a dozen prim and thrifty boroughs for one such lovely greatheart!' Too many chocolates and you start to feel sick.
With Gavin Young, whose Slow Boats Home (Hutchinson hardback pounds 12.95 pp441) is the plump sequel to his equally plump Slow Boats to China (Penguin pounds 3.95 pp512), the diet is less rich and does not pall. Young is both a different kind of writer from Morris and a different breed of traveler. Where she whizzes from one city to the next, he stays right away from them. Islands are what he likes, and ships, floating islands, and those who sail on them. The style is baggier, with more time for other people's sense of the world.
From a bath house in Hong Kong – the only city he dwells on – to a hut on Cape Horn at the height of the Falklands War Young finds people who speak freely, even when he can barely understand what they say. His journeys are animated not so much by change of landscape as by the sound of voices, songs, arguments, whispered invitations, the buzz of mosquitoes, the sigh of wind in the rigging. Slow Boats Home is a truly charming book; from the tiny islands of Oceania, paradises newly lost to drink, drugs, violence and videos, to South America, St Helena and homecoming to an England that seems to be no longer truly home.
Get Away From It All To The Falklands
Between the glossy pages devoted to pony trekking in Outer Mongolia and bird-watching in Galapagos Islands, travel brochures will soon be singing to their richer readers the siren song of a new tourist destination – the Falklands.
Let's face it. The only reason people know about the Falklands is because a war was fought there. But the Falklands are actually a beautiful place to visit. So push thoughts of war out of your mind and replace them with images of lush, spectacular scenery.
The Falkland Islands Development Corporation, based in Port Stanley, is going ahead with plans to establish a modest tourist industry which it hopes will attract 1,000 visitors a year and contribute an annual pounds 500,000 to the islands' precarious economy.
By the end of this year the corporation hopes to have completed the first of a chain of 10-roomed mini-hotels on the islands' premier wildlife sites, to insulate European and North American visitors from the worst rigours of the sometimes-spartan Falklands lifestyle. Mr Simon Armstrong, the corporation's general manager, said in London yesterday: 'We aim to offer them ice in their whisky and a hot bath to drink it in.'
Holidays will be sold on the attractions of the islands' rich wildlife, including penguin, albatross, elephant seal and the unique Falklands fightless steamer duck. There are no plans to emphasize, or to offer tours of, the battlefields of the 1982 conflict.
Within the next few days the corporation hopes to be able to announce that the Ministry of Defence has agreed to extend concessionary fares to the Falklands to bona fide tourists. At present these fares are limited to islanders and others on essential business. The cost of return ticket from RAF Brize Norton to the new airport at Mount Pleasant, at present the only way of getting there, would then be reduced from the normal pounds 2,250 to about pounds 1,050.
Two-week package holidays could then be priced at about pounds 2,000, including return flight, internal air travel and accommodation.
The corporation admits that selling the islands' tourist potential would be much easier if air links could be re-established with South America; exploratory talks are being held with Chilean airlines to examine the political and economic feasibility of a regular link with Chile.
Tourists expecting the occasional relief of a night on the tiles will, however, find that the island capital is still seriously deficient in after-dark amusement. But that will probably change over the years as the Falklands become a more popular tourist destination.
Adventures in Bicycling
Paris Removes Her Make Up
Travel: The Last Wilderness of Arctic Europe
Real-Life Television Drama in the Border Country
Illness on Increase as People Fail to Take More Exercise
© Copyright 2016. All Rights Reserved by Webriti


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\section{Concluding Remarks}
In this paper, we studied the problem of exact partitioning of high-order planted models. We proposed a novel convex tensor optimization problem which, given the sufficient condition in Theorem \ref{thm:main} is satisfied, recovers the underlying cluster structure perfectly with probability tending to $1$.

A key motivation behind our work is to develop efficient exact partitioning algorithms for high-order planted models with provable guarantees. To the best of our knowledge the approach is novel for hypergraph models. There have also been some works taking some more conventional approaches, for example, information-theoretic bounds in densest subhypergraphs \citep{buhmannrecovery} and hypergraph SBMs \citep{corinzia2019exact}, as well as detection (i.e., determining whether there exists certain planted structures) in hypergraph SBMs \citep{angelini2015spectral} and hypergraph planted cliques \citep{zhang2018tensor}. These could be a direction in our future works.

\section{Introduction}
On a higher level, a \emph{planted model} simulates interactions among various groups of entities in a network. Typical planted models assume that nodes are grouped into a number of clusters, and each pair of nodes is connected randomly with some probability related to the cluster membership. The generative and non-deterministic nature makes planted models of both theoretical and practical interests in the field of community detection, data mining, engineering, biology, among others. Various classical planted models have been studied extensively in recent years.
This includes, for instance, the stochastic block models (SBMs) \citep{mossel2016belief,abbe2015exact,abbe2015community,amini2018semidefinite}, the planted cliques \citep{barak2019nearly, ames2011nuclear}, the densest subgraphs \citep{fang2019efficient, arias2014community}, the latent space models \citep{chen2020near, ke2018information}.

However in more complex real-world systems of interest, entities may interact beyond the pairwise way. One example is the folksonomy, where a group of individuals collaboratively annotate a data set to create semantic structure \citep{ghoshal2009random}. Such networks usually exhibit a "user–resource–annotation" tripartite structure, and naturally can be modeled as a 3-uniform hypergraph \citep{ghoshdastidar2017consistency}. In the past decades researchers have been utilizing hypergraphs to model a number of real-life networks, for instance, brain regions \citep{gu2017functional,zu2016identifying}, food webs \citep{li2017inhomogeneous}, images \citep{li2013contextual}, VLSI designs \citep{karypis1999multilevel}. Hypergraphs are the generalization of ordinary (pairwise) graphs, where each hyperedge connects multiple nodes. By introducing hyperedges one may capture potential higher-order interactions among nodes, which are rather prevalent in the aforementioned real-life tasks.

The problem of partitioning high-order models have been studied for some long time. Graph-theoretic problems including cuts, colorings, and traversals have been analyzed in earlier works \citep{berge1984hypergraphs, karypis1999multilevel}. In the past decade, researchers have started looking at spectral theory and algebraic connectivity of hypergraphs \citep{pearson2014spectral,cooper2012spectra,hu2012algebraic, zhou2007learning}.
Certain high-order models of interests have received more attention from an algorithmic point of view. This includes the densest subhypergraphs \citep{buhmannrecovery, chlamtac2016densest, taylor2016approximations}, the hypergraph SBMs \citep{kim2017community, florescu2016spectral}, and the hypergraph planted cliques \citep{zhang2018tensor}.
Despite years of research, however, very little is known about the efficient \emph{exact partitioning} conditions in general high-order planted models.

In this paper we propose an efficient convex optimization approach for exact partitioning of high-order planted models. A high-order planted model generates hypergraphs, which simulate multi-entity interactions in a network. Our model class formulation is highly general and subsumes models analyzed and applied in prior literature, including the disjoint hypercliques, the densest subhypergraphs, and the hypergraph stochastic block models. When the order is set to $2$, our definition of high-order planted models reduces to regular planted models with ordinary graphs.

It is known that with certain constraints such as balancedness, the problem of computing an optimal partitioning of a hypergraph is NP-hard in general \citep{BORNDORFER201515, lyaudet2010np}. In spite of being computationally hard, we provide an efficient exact partitioning algorithm, which recovers the underlying cluster structures with probability tending to $1$ if certain statistical conditions are fulfilled. Hypergraph partitioning algorithms in prior literature either are objective function approximation algorithms \citep{chlamtac2016densest,taylor2016approximations}, or unfold hypergraphs into matrices \citep{kim2017community, ghoshdastidar2017consistency}. On the contrary by ``exact partitioning'', our algorithm returns the true solution perfectly (up to the permutation of clusters) using a novel tensor optimization approach. Our exact partitioning algorithm is formulated as a convex optimization problem, which can be solved efficiently using interior point methods \citep{boyd2004convex}.

Motivated by the use of adjacency matrices in ordinary graphs, researchers have been using tensors, or multidimensional arrays, to embed the information in hypergraphs. When dealing with tensors, one common approach is to unfold the tensor into matrices, and apply partitioning algorithms for ordinary graphs \citep{lu2019tensor, zhou2007learning}. There also exists some prior works using Sum-of-Squares (SoS) based relaxations \citep{kim2017community}. In this paper however, we let tensors be tensors; we are interested in generic tensor methods. Our proof relies on a careful construction of tensor projections and a novel analysis of tensor nuclear norm constraints. It is worth mentioning that tensor nuclear norms have been used extensively in problems related to tensor completion \citep{yuan2017incoherent, yuan2016tensor}. To the best of our knowledge, the use of tensor nuclear norm in hypergraph partitioning problems is novel.

The feasibility of efficient exact partitioning depends on the signal-to-noise ratio (SNR) in the planted model. In our model the SNR is determined by two signal parameters $p$ and $q$ (see Definition \ref{def:model} for details). We show that in high-order planted models, the problem becomes statistically easier if the gap between $p$ and $q$ becomes larger. Intuitively, a larger SNR (i.e., larger gap between $p$ and $q$) leads to cleaner cluster structures in the observed hypergraph and the adjacency tensor. The generative nature enables us to study the average-case behaviors of high-order planted models. Our analysis establishes the regime in which efficient exact recovery of hidden cluster structures is possible from noisy observation of hypergraphs.

\textbf{Related Works.} There has been a lot of research on the partitioning of certain high-order planted models.
For the densest subhypergraphs, \citet{chlamtac2016densest} and \citet{taylor2016approximations} proposed theoretical objective function approximation algorithms. Our goal, arguably more challenging, is to recover the true underlying clustering structure. For hypergraph SBMs, approaches include truncating the hypergraph to a multigraph \citep{kim2017community} or an ordinary graph with a weighted adjacency matrix \citep{ghoshdastidar2017consistency}.
It is worth highlighting that our result is not merely an extension. Our definition of high-order planted models is highly general, and the convex optimization formulation with a tensor nuclear norm constraint is novel. Moreover our approach does not approximate, or truncates the hypergraph to an ordinary graph. To the best of authors' knowledge, we are the first one applying tensor nuclear norm methods in high-order partitioning problems.

\textbf{Summary of our contributions.} We provide a series of novel results in this paper:
\begin{itemize}
    \item We propose the highly general class definition of high-order planted models, and demonstrate that our model class definition subsumes several existing planted models, including the disjoint hypercliques, the densest subhypergraphs, and the hypergraph stochastic block models.
    \item We formulate the problem of exact partitioning in high-order planted models as a novel tensor optimization problem with a tensor nuclear norm constraint, and we establish the regime in which hidden cluster structures can be recovered efficiently.
\end{itemize}

\section{Preliminaries}
\subsection{Notations}
\label{section:notation}
In this section, we introduce the notations that will be used in the paper.

We use lowercase font (e.g., $a,b,u,v$) for scalars and vectors, uppercase font (e.g., $A, B, C$) for matrices, and calligraphic font (e.g., $\ten{A}, \ten{B}, \ten{C}$) for tensors. We use ${\mathbb{R}}$ to denote the set of real numbers.

For any integer $n$, we use $[n]$ to denote the set $\{1, \ldots, n\}$.
For clarity when dealing with a sequence of objects, we use the superscript ${(i)}$ to denote the $i$-th object in the sequence, and subscript $j$ to denote the $j$-th entry. For example, for a sequence of vectors $\{x^{(i)}\}_{i \in [n]}$, $x^{(1)}_2$ represents the second entry of vector $x^{(1)}$. The notation $\otimes$ is used to denote outer product of vectors, for example, $x^{(1)} \otimes \ldots \otimes x^{(m)}$ is a tensor of order $m$, such that
\begin{align*}
(x^{(1)} \otimes \ldots \otimes x^{(m)})_{i_1, \ldots, i_m} := x^{(1)}_{i_1} \ldots x^{(m)}_{i_m} \,.
\end{align*}
We use $\textbf{1}$ to denote the all-one vector.

Let $\ten{A} = (\ten{A}_{i_1, \dots, i_m})$ be an $m$-order tensor of size ${n_1 \times \cdots \times n_m}$.
Tensor $\ten{A}$ is \emph{symmetric} if it is invariant under any permutation of its indices, i.e.,
\[
\ten{A}_{\sigma(i_1),\ldots, \sigma(i_m)} = \ten{A}_{i_1,\ldots, i_m} \,,
\]
for any permutation $\sigma: [m] \to [m]$.

For tensor $\ten{A} = (\ten{A}_{i_1, \dots, i_m})$ and $\ten{B} = (\ten{B}_{i_1, \dots, i_m})$ of same size, we define the \emph{inner product} of $\ten{A}$ and $\ten{B}$ as
\[
\inprod[\ten{A}]{\ten{B}} := \sum_{i_1,\ldots,i_m} \ten{A}_{i_1 ,\ldots, i_m} \ten{B}_{i_1 ,\ldots, i_m} \,.
\]

Tensor addition and subtraction are defined entrywise, e.g., $\ten{A} + \ten{B} = (\ten{A}_{i_1, \dots, i_m} + \ten{B}_{i_1, \dots, i_m})$. With slight abuse of notation, for any constant $c \in {\mathbb{R}}$, we use $\ten{A} < c$ to denote entrywise inequality.

For any vector $u \in {\mathbb{R}}^n$, we denote the corresponding $m$-th order $\emph{rank-one}$ tensor as $u^{{\otimes m}}$, where
\[
(u^{{\otimes m}})_{i_1 ,\ldots, i_m} := u_{i_1} \ldots u_{i_m} \,.
\]

For any symmetric tensor $\ten{A}$, we define its \emph{spectral norm} as
\[
\tnorm{\ten{A}} := \sup_{u \in {\mathbb{S}^{n-1}}} \abs{\inprod[\ten{A}]{u^{\otimes m}}} \,,
\]
where ${\mathbb{S}^{n-1}}$ denotes the unit sphere. Similarly we define its \emph{nuclear norm} as
\[
\tnormnu{\ten{A}} := \inf \left\{ \sum_{i=1}^r \abs{\lambda_i} : A = \sum_{i=1}^r \lambda_i u^{(i) {\otimes m}}, u^{(i)} \in {\mathbb{S}^{n-1}}\right\} \,.
\]
It is worth mentioning that, like the Schatten $p$-norms in the matrix case, the tensor spectral norm and tensor nuclear norm are also dual to each other \citep{friedland2018nuclear}.

Regarding entrywise norms, we define the tensor $L_1$ norm and the tensor $L_\infty$ norm of $\ten{A}$ respectively as
\[
\tnormone{\ten{A}} := \sum_{i_1,\dots,i_m} \abs{\ten{A}_{i_1,\dots,i_m}} \,, \qquad
\tnorminf{\ten{A}} := \max_{i_1,\dots,i_m} \abs{\ten{A}_{i_1,\dots,i_m}} \,.
\]

A tensor \emph{fiber} is a vector obtained by holding all but one mode constant. For example, using MATLAB-style notations, vector $\ten{A}(i_1, \dots, i_{j-1}, :, i_{j+1}, \dots, i_m)$ is a mode-$j$ fiber of $\ten{A}$.
Define
\[
{\mathcal{L}}_j (\ten{A}) := \spa{\{\ten{A}(i_1, \dots, i_{j-1}, :, i_{j+1}, \dots, i_m)\}_{i_k \in [n_k]}}
\]
as the vector space spanned by all mode-$j$ fibers of $\ten{A}$.

In this paper we will frequently use tensor projection operators in our analysis. We use $\mathcal{I}: {\mathbb{R}}^n \to {\mathbb{R}}^n$ to denote the identity projection. For any tensor $\ten{A}$ of size $n_1 \times \dots \times n_m$, define $\bm{P}^j_\ten{A}: {\mathbb{R}}^{n_j} \to {\mathcal{L}}_j (\ten{A})$ as the orthogonal projection to ${\mathcal{L}}_j (\ten{A})$. Similarly we define $\bm{P}^j_{\ten{A}^\perp} = \mathcal{I} - \bm{P}^j_\ten{A}$ to be the projection to the orthogonal complement ${\mathcal{L}}_j^\perp (\ten{A})$ of ${\mathcal{L}}_j (\ten{A})$.
We define the tensor projection with respect to $\ten{A}$ as follows
\begin{align*}
\mathcal{Q}_\ten{A}^0 &:= \bm{P}_\ten{A}^1 \otimes \dots \otimes \bm{P}_\ten{A}^m , \qquad
\mathcal{Q}_\ten{A}^i := \bm{P}_\ten{A}^1 \otimes \dots \otimes \bm{P}_\ten{A}^{i-1} \otimes \bm{P}_{\ten{A}^\perp}^{i} \otimes \bm{P}_\ten{A}^{i+1} \otimes \dots \otimes \bm{P}_\ten{A}^m
\end{align*}
for $i \in [m]$, and
\begin{align*}
\mathcal{Q}_\ten{A} := \sum_{j=0}^m \mathcal{Q}_\ten{A}^j, \qquad
\mathcal{Q}_{\ten{A}^\perp} := \mathcal{I} - \mathcal{Q}_\ten{A} \,.
\end{align*}

\subsection{High-order Planted Models}
\label{section:def_model}

We now introduce the definition of \emph{high-order planted models}.
\begin{definition}[High-order Planted Models]
A high-order planted model is denoted as $\mathcal{M}(n,m,r,k,p,q)$, where $n$ is the number of vertices, and $m$ is the order of the model. It is assumed that uniformly at random, $rk$ out of $n$ vertices are grouped into $r$ clusters of equal size $k$, and the remaining $n - rk$ vertices do not belong any cluster. $p, q$ are signal parameters satisfying $0 \leq q < p \leq 1$.

Model $\mathcal{M}(n,m,r,k,p,q)$ generates a random hypergraph $\mathcal{G} = (\mathcal{V},\mathcal{E})$ in the following way. For each $m$-tuple $(v_{i_1},\ldots,v_{i_m}) \subset \mathcal{V}$, if all $m$ vertices are from the same cluster, nature adds the hyperedge $(v_{i_1},\ldots,v_{i_m})$ to $\mathcal{E}$ with probability $p$; otherwise, nature adds the hyperedge with probability $q$.
\label{def:model}
\end{definition}

Our goal is to recover the cluster membership of vertices in model $\mathcal{M}$ from the observed hypergraph $\mathcal{G}$. For any $i\in[r]$, we use $\vct{y}^{(i) \ast} \in \{0,1\}^n$ to denote the true membership vector of cluster $i$, such that  $\vct{y}^{(i) \ast}_j = 1$ if vertex $j$ is in cluster $i$, and $0$ otherwise.
We introduce the agreement tensor ${\ten{Y}^\ast} = \sum_{i=1}^r \vct{y}^{(i)\ast{\otimes m}}$. It is not hard to see that ${\ten{Y}^\ast}$ is $0$-$1$ valued, as the clusters are non-overlapping. Thus, ${\ten{Y}^\ast}$ encodes all cluster membership information (up to the permutation of clusters, as there is no way to distinguish between clusters without prior knowledge).
Let $\ten{A}$ be the adjacency tensor of hypergraph $\mathcal{G}$. From the definition above, $\ten{A}$ is a symmetric tensor. Each entry in $\ten{A}$ is generated to be $1$ with probability $p$ if the corresponding entry in ${\ten{Y}^\ast}$ is $1$; otherwise it is generated to be $1$ with probability $q$. The problem now reduces to recover ${\ten{Y}^\ast}$ from the observation of $\ten{A}$.

\textbf{Classical models.} Here are some classical models covered by our definition.
\begin{itemize}
    \item \textbf{Disjoint Hypercliques}: $p = 1, 0<q<1$. In this case, $r$ hypercliques of size $k$ are planted in the observed hypergraph $\mathcal{G}$. A hyperclique is the generalization of graph cliques. In a hyperclique, every distinct $m$-tuple is connected by a hyperedge \citep{nie2017symmetric}.
    \item \textbf{Densest Subhypergraph}: $0<q<p<1, r=1$. In this case there exists a dense subhypergraph of size $k$ in the observed hypergraph $\mathcal{G}$.
    \item \textbf{Hypergraph Stochastic Block Model}: $0<q<p<1, n = rk, r\geq 2$. In this case there exists $r$ dense subhypergraphs of size $k$ in the observed hypergraph $\mathcal{G}$.
\end{itemize}

\textbf{Remark on diagonal entries.} One can see that in the adjacency tensor $\ten{A}$, most entries follow Bernoulli distribution with parameter $p$ or $q$. A natural question is: what about the diagonal entries? By ``diagonal'', we refer to the entries with at least one duplicate index, for example, $\ten{A}_{1,1,3,4}$ in a $4$th order model. Mathematically, we are referring to the set $\lbrace \ten{A}_{i_1,\dots, i_m} : \abs{\{   i_1,\dots,i_m    \}} < m\rbrace$. \begin{itemize}
    \item A trivial approach is to force all diagonal entries to be $0$. This is customary in the stochastic block model \citep{abbe2015exact, chen2014statistical}. In the context of graph theory, this means that $\mathcal{G}$ is an \emph{$m$-uniform} hypergraph, where all hyperedges have size $m$.
    \item Another approach is to allow the diagonal entries to be Bernoulli random variables with parameter $p$ or $q$, depending on the corresponding entries in ${\ten{Y}^\ast}$. In this case the hypergraph $\mathcal{G}$ is no longer required to be uniform.
\end{itemize}
In the following analysis we adopt the latter approach for its generality. We want to highlight that the technical difference is marginal, however. Either choice will not break the framework of our analysis.
\section{Efficient Exact Partitioning}

In this section, we propose and analyze an algorithm which efficiently recovers the true underlying cluster structures in high-order planted models.
Recall that ${\ten{Y}^\ast}$ is the true agreement tensor. We say an algorithm achieves \emph{exact partitioning}, if its output $\ten{Y}$ is identical to ${\ten{Y}^\ast}$.

 \begin{algorithm}
    \caption{Exact Partitioning of High-order Models}\label{euclid}
    \hspace*{\algorithmicindent} \textbf{Input:} adjacency tensor $\ten{A}$\\
    \hspace*{\algorithmicindent} \textbf{Output:} estimated agreement tensor $\ten{Y}$
\begin{align}
\operatorname*{maximize}_{\ten{Y}} \qquad & \langle \ten{A}, \ten{Y}\rangle  \nonumber\\
\operatorname*{subject\; to} \qquad
&\tnormnu{\ten{Y}} \leq rk^{m/2} \nonumber\\
&\inprod[\bm{1}^{\otimes m}]{\ten{Y}} = rk^m \nonumber\\
&0 \leq \ten{Y} \leq 1\,,
\label{opt:primal}
\end{align}
\end{algorithm}

In the following analysis we examine the statistical conditions for problem \eqref{opt:primal} to succeed with high probability. Note that the objective function and constraints in problem \eqref{opt:primal} are convex. It is known that convex optimization problems can be solved efficiently using interior point methods \citep{boyd2004convex}. \textbf{Our analysis establishes the regime in which given the adjacency tensor $\ten{A}$, the true underlying cluster structures ${\ten{Y}^\ast}$ can be recovered by problem \eqref{opt:primal} efficiently and perfectly.}

\textbf{Remark on exact partitioning.} It is worth mentioning that our algorithm does not require any rounding step and outputs the exact solution. Hypergraph partitioning algorithms in prior literature either are objective function approximation algorithms \citep{chlamtac2016densest,taylor2016approximations}, or unfold hypergraphs into matrices \citep{kim2017community, ghoshdastidar2017consistency}. Note that the groundtruth ${\ten{Y}^\ast}$ is a feasible solution to problem \eqref{opt:primal}. Our analysis states that if certain statistical conditions are satisfied, with high probability no other feasible solution $\ten{Y} \neq {\ten{Y}^\ast}$ will achieve a better objective value.

We now present the main theorem, which provides a sufficient condition for problem \eqref{opt:primal} to succeed with high probability.
\begin{theorem}
Consider any hypergraph $\mathcal{G}$ sampled from a high-order model $\mathcal{M}(n,m,r,k,p,q)$. Let $\ten{A}$ be the adjacency tensor of $\mathcal{G}$. If
\begin{equation}
\frac{(p-q)^2}{p(1-q)} = \Omega\left( \frac{nm^5 \log m}{k^{m-1}} \right) \,,
\label{eq:thm}
\end{equation}
then problem \eqref{opt:primal} recovers the underlying cluster structure of $\mathcal{M}$ perfectly with probability at least $1-O(1/n)$.
\label{thm:main}
\end{theorem}

\textbf{Remark on rates.} In high-order planted models, $p$ and $q$ are signal parameters that determine the signal-to-noise ratio (SNR) of the model. This is implied by the left-hand side of \eqref{eq:thm}: as the gap $p-q$ becomes larger, SNR becomes higher and exact partitioning gets easier. On the right-hand side, one can notice that as $n$ gets larger, the whole term becomes smaller (remember that $n = rk$). From an information-theoretical point of view this is intuitive, as a larger number of samples leads to easier recovery of the true signal.

It would also be interesting to compare our rates with those of ordinary planted models.
In the case of $m=2$, our condition becomes
$(p-q)^2 k^2 \geq Cp(1-q) k n$ for some constant $C$, while the condition in \citet{chen2014statistical} is $(p-q)^2 k^2 \geq C(p(1-q)k\log n + q(1-q)n )$. Comparison on the right-hand sides shows that our bound only requires a slightly higher order ($\Omega(kn)$ versus $\Omega(k\log n + n)$).

\subsection{Technical Lemmas}
We first present some technical lemmas that will be used to prove our main result in Theorem \ref{thm:main}.
\begin{lemma}
Let $\ten{A}$ be an $m$-th order symmetric random tensor of size ${\mathbb{R}}^{n \times \dots \times n}$. Assume that each entry of $\ten{A}$ is independent and follows Bernoulli distribution with parameter $p$ or $q$, such that $0\leq q<p \leq 1$. Then with probability at least $1- O(1/n)$, we have
\begin{equation}
\tnorm{\ten{A} - \Expect{\ten{A}}}
\leq
C \sqrt{p(1-q) mn\log m } \,,
\end{equation}
for some large enough constant $C$.
\label{lemma:concen_A}
\end{lemma}
\begin{proof}
First note that the entries in $\ten{A} - \Expect{\ten{A}}$ are independent (up to symmetry) and zero-mean. Furthermore, the variance of each entry is bounded above by $p(1-q)$. Then \citet[Lemma 1, Theorem 1]{tomioka2014spectral} implies
\begin{align*}
\tnorm{\ten{A} - \Expect{\ten{A}}}
\leq
\sqrt{8 p(1-q) \left(mn\log(5m) + \log (2/\delta)\right)}
\end{align*}
with probability at least $1-\delta$. Setting $\delta$ to be $O(1/n)$ leads to
\begin{align*}
\tnorm{\ten{A} - \Expect{\ten{A}}}
\leq
C \sqrt{ p(1-q) mn\log m }
\end{align*}
with probability at least $1- O(1/n)$, for some large enough constant $C$.
\end{proof}

\begin{lemma}
For any $m$-th order tensor $\ten{X} \in {\mathbb{R}}^{n_1 \times \dots \times n_m}$, there exists a $\ten{W}_0 \in {\mathbb{R}}^{n_1 \times \dots \times n_m}$ such that
\[
\ten{W}_0 = \mathcal{Q}_\ten{X}^0 (\ten{W}_0), \quad \tnorm{\ten{W}_0} = 1, \quad \text{and } \tnormnu{\ten{X}} = \inprod[\ten{W}_0]{\ten{X}} \,.
\]

Furthermore, for any $\ten{X}', \ten{W}_1 \in {\mathbb{R}}^{n_1 \times \dots \times n_m}$ with $\tnorm{\ten{W}_1} \leq \frac{2}{m(m-1)}$,
\begin{equation}
\tnormnu{\ten{X}'} \geq \tnormnu{\ten{X}} + \inprod[\ten{W}_0 + \mathcal{Q}_{\ten{X}^\perp} (\ten{W}_1)]{\ten{X}' - \ten{X}} \,.
\end{equation}
\label{lemma:yuan2017}
\end{lemma}
\begin{proof}
The first claim follows the proof of \citet[Theorem 1]{yuan2017incoherent} by setting $\delta = 1$. It is also proved that any $\ten{W}_1 \in {\mathbb{R}}^{n_1 \times \dots \times n_m}$ such that $\tnorm{\ten{W}_1} \leq \frac{2}{m(m-1)}$, one has
$
\tnorm{\ten{W}_0 + \mathcal{Q}_{\ten{X}^\perp} (\ten{W}_1)} \leq 1 \,.
$
Then, since the tensor spectral norm and tensor nuclear norm are dual to each other, and by orthogonality of projection, it follows that
\begin{align*}
\inprod[\ten{W}_0 + \mathcal{Q}_{\ten{X}^\perp} (\ten{W}_1)]{\ten{X}' - \ten{X}}
&= \inprod[\ten{W}_0 + \mathcal{Q}_{\ten{X}^\perp} (\ten{W}_1)]{\ten{X}'} - \inprod[\ten{W}_0 + \mathcal{Q}_{\ten{X}^\perp} (\ten{W}_1)]{\ten{X}} \\
&\leq \tnorm{\ten{W}_0 + \mathcal{Q}_{\ten{X}^\perp} (\ten{W}_1)} \cdot \tnormnu{\ten{X}'} - \inprod[\ten{W}_0]{\ten{X}} \\
&\leq \tnormnu{\ten{X}'} - \inprod[\ten{W}_0]{\ten{X}} \\
&= \tnormnu{\ten{X}'} - \tnormnu{\ten{X}} \,.
\end{align*}
This completes the proof of the second claim.
\end{proof}

\subsection{Proof of Theorem \ref{thm:main}}
Armed by the technical lemmas above, we now focus on the proof of our main result in Theorem \ref{thm:main}.

\begin{proof}
In the following proofs we use ${\ten{Y}^\ast}$ to denote the true agreement tensor as defined in Section \ref{section:def_model}. We also define $\Delta(\ten{Y}) = \inprod[\ten{A}]{\ty^\ast - \ty}$. For any vertex $i\in [n]$, we use $N(i) := \{j : i,j \text{ are in the same cluster}\}$ to denote the neighborhood of $i$. Note that $\abs{N(i)} = k$.

To prove ${\ten{Y}^\ast}$ is the optimal solution to problem \eqref{opt:primal}, our goal is to prove $\Delta(\ten{Y}) > 0$ for every feasible $\ten{Y} \neq \ten{Y}^\ast$ satisfying the constraints in problem \eqref{opt:primal}, with high probability.
It is worth mentioning that the groundtruth ${\ten{Y}^\ast}$ is always a feasible solution to problem \eqref{opt:primal}, with $\tnormnu{{\ten{Y}^\ast}} = rk^{m/2}$.

To start the analysis we first split $\Delta(\ten{Y})$ as follows
\begin{equation}
\Delta(\ten{Y}) = \inprod[\Expect{\ten{A}}]{\ty^\ast - \ty} + \inprod[\ten{A} - \Expect{\ten{A}}]{\ty^\ast - \ty}\,.
\label{eq:dy_twoparts}
\end{equation}
We now proceed to characterize these two terms.
Recall that $\ten{A}$ is an $m$-th order symmetric random tensor of size ${\mathbb{R}}^{n \times \dots \times n}$, and each entry of $\ten{A}$ is independent and follows Bernoulli distribution with parameter $p$ or $q$.

For the first term in $\eqref{eq:dy_twoparts}$, we have
\begin{align}
\inprod[\Expect{\ten{A}}]{\ty^\ast - \ty}
&= \inprod[q\bm{1}^{\otimes m} + (p-q){\ten{Y}^\ast}]{\ty^\ast - \ty} \nonumber\\
&= q\inprod[\bm{1}^{\otimes m}]{\ty^\ast - \ty} + (p-q)\inprod[{\ten{Y}^\ast}]{\ty^\ast - \ty} \nonumber\\
&= 0 + (p-q) \sum_{i_1,\dots,i_m} \textbf{1}[{{\ten{Y}^\ast}_{i_1,\dots,i_m} = 1} \land {\ten{Y}_{i_1,\dots,i_m} = 0}] \nonumber\\
&= \frac{1}{2}(p-q)\tnormone{\ty^\ast - \ty} \,,
\label{eq:dy_expectation}
\end{align}
where the last two equalities hold because $\inprod[\bm{1}^{\otimes m}]{\ten{Y}} = \inprod[\bm{1}^{\otimes m}]{{\ten{Y}^\ast}} = rk^m$ and $0 \leq \ten{Y} \leq 1$.

To bound second term in $\eqref{eq:dy_twoparts}$, we first rewrite $\inprod[\ten{A} - \Expect{\ten{A}}]{\ty^\ast - \ty}$ as $\lambda \inprod[\ten{Z}]{\ty^\ast - \ty}$, where $\lambda := \frac{m(m-1)}{2} \cdot C \sqrt{ p(1-q) mn\log m }$. Lemma \ref{lemma:concen_A} guarantees $\tnorm{\ten{Z}} \leq \frac{2}{m(m-1)}$ with high probability.

Before we continue bounding $\lambda \inprod[\ten{Z}]{\ty^\ast - \ty}$, here we take a pause, and instead characterize the subdifferential of $\tnormnu{{\ten{Y}^\ast}}$.
Note that for any $\ten{Y}$ and ${\ten{Y}^\ast}$, from the definition of tensor spectral norm we have $\tnorm{\ten{Y}} = \tnorm{{\ten{Y}^\ast}} = k^{m/2}$.
Setting $\ten{X} = {\ten{Y}^\ast}$  in Lemma \ref{lemma:yuan2017}, it can be verified that the conditions
\[
\ten{W}_0 = \mathcal{Q}_{{\ten{Y}^\ast}}^0 (\ten{W}_0), \quad \tnorm{\ten{W}_0} = 1, \quad \tnormnu{{\ten{Y}^\ast}} = \inprod[\ten{W}_0]{{\ten{Y}^\ast}}
\]
holds for the choice $\ten{W}_0 = k^{-m/2}{\ten{Y}^\ast}$.
Furthermore, by setting $\ten{X}' = \ten{Y}$ and $\ten{W}_1 = \ten{Z}$ , it follows that
\begin{equation}
\tnormnu{\ten{Y}}
\geq
\tnormnu{{\ten{Y}^\ast}} + \inprod[k^{-m/2}{\ten{Y}^\ast} + \mathcal{Q}_{\ten{Y}^{\ast\perp}} (\ten{Z})]{\ten{Y} - {\ten{Y}^\ast}} \,.
\label{eq:subdifferential}
\end{equation}
In addition, problem \eqref{opt:primal} requires $\tnormnu{\ten{Y}} \leq rk^{m/2}$ for every $\ten{Y}$. Since $\tnormnu{{\ten{Y}^\ast}} = rk^{m/2}$, we obtain
$\tnormnu{\ten{Y}} - \tnormnu{{\ten{Y}^\ast}} \leq 0$ for every feasible $\ten{Y}$. Combining this with \eqref{eq:subdifferential} we obtain
\begin{equation}
\inprod[k^{-m/2}{\ten{Y}^\ast} + \mathcal{Q}_{\ten{Y}^{\ast\perp}} (\ten{Z})]{\ty^\ast - \ty}
\geq
0 \,.
\label{eq:subgeq0}
\end{equation}

We can now continue to bound
$\lambda \inprod[\ten{Z}]{\ty^\ast - \ty}$.
It follows that
\begin{align}
\lambda \inprod[\ten{Z}]{\ty^\ast - \ty}
&= \lambda \inprod[\mathcal{Q}_{{\ten{Y}^\ast}} (\ten{Z}) + \mathcal{Q}_{\ten{Y}^{\ast\perp}} (\ten{Z})]{\ty^\ast - \ty} \nonumber\\
&\geq \lambda \inprod[\mathcal{Q}_{{\ten{Y}^\ast}} (\ten{Z}) + \mathcal{Q}_{\ten{Y}^{\ast\perp}} (\ten{Z})]{\ty^\ast - \ty} - \lambda \inprod[k^{-m/2}{\ten{Y}^\ast} + \mathcal{Q}_{\ten{Y}^{\ast\perp}}(\ten{Z})]{\ty^\ast - \ty} \nonumber\\
&= \lambda \inprod[\mathcal{Q}_{\ten{Y}^\ast} (\ten{Z}) - k^{-m/2}{\ten{Y}^\ast}]{\ty^\ast - \ty} \,,
\label{eq:dy_deviation}
\end{align}
where the inequality holds by introducing \eqref{eq:subgeq0}.

Plugging the results from \eqref{eq:dy_expectation} and \eqref{eq:dy_deviation} into \eqref{eq:dy_twoparts}, since the tensor $L_1$ norm and tensor $L_\infty$ norm are dual to each other, we obtain
\begin{align}
\Delta(\ten{Y})
&\geq \frac{1}{2}(p-q)\tnormone{\ty^\ast - \ty} + \lambda \inprod[\mathcal{Q}_{\ten{Y}^\ast} (\ten{Z}) - k^{-m/2}{\ten{Y}^\ast}]{\ty^\ast - \ty} \nonumber\\
&\geq \left( \frac{1}{2}(p-q) - \lambda k^{-m/2}\tnorminf{{\ten{Y}^\ast}} - \lambda\tnorminf{\mathcal{Q}_{\ten{Y}^\ast} (\ten{Z})} \right) \tnormone{\ty^\ast - \ty} \nonumber\\
&= \left( \frac{1}{2}(p-q) - \lambda k^{-m/2} - \tnorminf{\mathcal{Q}_{\ten{Y}^\ast} (\lambda\ten{Z})} \right) \tnormone{\ty^\ast - \ty} \,.
\end{align}
Since $\tnormone{\ty^\ast - \ty}$ is always positive, $\Delta(\ten{Y}) \geq 0$ holds if
\begin{equation}
\frac{1}{2}(p-q) - \lambda k^{-m/2} - \tnorminf{\mathcal{Q}_{\ten{Y}^\ast} (\lambda\ten{Z})}
\geq 0 \,.
\label{eq:suff_condition}
\end{equation}

We now want to bound the tensor $L_\infty$ norm $\tnorminf{\mathcal{Q}_{\ten{Y}^\ast} (\lambda \ten{Z})}$ in \eqref{eq:suff_condition}. Since ${\ten{Y}^\ast}$ is a symmetric tensor, the linear spaces spanned by fibers along each mode are identical, i.e., ${\mathcal{L}}_1({\ten{Y}^\ast}) = \dots = {\mathcal{L}}_m({\ten{Y}^\ast}) = {\mathcal{L}}({\ten{Y}^\ast})$. It follows that $\bm{P}_{{\ten{Y}^\ast}}^1 = \dots = \bm{P}_{{\ten{Y}^\ast}}^m = \bm{P}_{{\ten{Y}^\ast}}$.
By definition of projection operators defined in Section \ref{section:notation}, it follows that
\begin{align*}
\mathcal{Q}_{\ten{Y}^\ast}
&= \sum_{j=0}^m \mathcal{Q}_{\ten{Y}^\ast}^j \\
&= (\mathcal{I} - \bm{P}_{\ten{Y}^\ast}) \otimes \bm{P}_{\ten{Y}^\ast} \otimes \dots \otimes \bm{P}_{\ten{Y}^\ast}
+ \bm{P}_{\ten{Y}^\ast} \otimes (\mathcal{I} - \bm{P}_{\ten{Y}^\ast}) \otimes \dots \otimes \bm{P}_{\ten{Y}^\ast} + \dots
 \\
&\quad + \bm{P}_{\ten{Y}^\ast} \otimes \dots \otimes (\mathcal{I} - \bm{P}_{\ten{Y}^\ast}) + \bm{P}_{\ten{Y}^\ast} \otimes \dots \otimes \bm{P}_{\ten{Y}^\ast} \\
&= \mathcal{I} \otimes \bm{P}_{\ten{Y}^\ast} \otimes \dots \otimes \bm{P}_{\ten{Y}^\ast}
+ \bm{P}_{\ten{Y}^\ast} \otimes \mathcal{I} \otimes \dots \otimes \bm{P}_{\ten{Y}^\ast} + \dots + \bm{P}_{\ten{Y}^\ast} \otimes \bm{P}_{\ten{Y}^\ast} \otimes \dots \otimes \mathcal{I} \\
&\quad- (m-1) \bm{P}_{\ten{Y}^\ast} \otimes \dots \otimes \bm{P}_{\ten{Y}^\ast} \,.
\end{align*}
Note that there are $m$ terms in the second to last line above. Due to symmetry and the fact that $\bm{P}_{\ten{Y}^\ast}$ is an orthogonal projection, we have
\[
\tnorminf{\mathcal{Q}_{\ten{Y}^\ast} (\lambda \ten{Z})} \leq (2m-1) \cdot \tnorminf{\mathcal{I} \otimes \bm{P}_{\ten{Y}^\ast} \otimes \dots \otimes \bm{P}_{\ten{Y}^\ast} (\lambda \ten{Z})} \,.
\]
We denote $\bar{\ten{A}} := \mathcal{I} \otimes \bm{P}_{\ten{Y}^\ast} \otimes \dots \otimes \bm{P}_{\ten{Y}^\ast} (\lambda \ten{Z})$ to be the projected tensor.
We now consider the linear space ${\mathcal{L}}({\ten{Y}^\ast})$. By definition of ${\ten{Y}^\ast}$, the spanned space has $r$ basis vectors $\vct{y}^{(1)\ast}, \dots, \vct{y}^{(r)\ast}$. Thus the orthogonal projection can be characterized as $\bm{P}_{\ten{Y}^\ast} = \frac{1}{k} \sum_{i=1}^r \vct{y}^{(i)\ast} \vct{y}^{(i)\ast\top} $.
To bound $\tnorminf{\mathcal{I} \otimes \bm{P}_{\ten{Y}^\ast} \otimes \dots \otimes \bm{P}_{\ten{Y}^\ast} (\lambda \ten{Z})} = \tnorminf{\bar{\ten{A}}}$, note that every single entry in $\bar{\ten{A}}$ is the average of $k^{m-1}$ independent random variables. Mathematically, we have
\begin{equation}
    \bar{\ten{A}}_{i, i_2, \dots, i_m} = \frac{1}{k^{m-1}}\sum_{j_2 \dots j_m \in N(i)} \lambda \ten{Z}_{i,j_2, \dots, j_m}
\label{eq:A_avg}
\end{equation}
for every $i,i_2,\dots,i_m$, where $N(i)$ is the neighborhood of $i$. Bernstein's inequality implies
\begin{equation}
\Prob{\sum_{j_2 \dots j_m \in N(i)} \lambda \ten{Z}_{i,j_2, \dots, j_m} \geq \sqrt{2(m+1)k^{m-1} p(1-q) \log n} + \frac{2}{3} (m+1)\log n}
\leq
n^{-(m+1)} \,.
\label{eq:bern_Z}
\end{equation}
Putting \eqref{eq:A_avg} and \eqref{eq:bern_Z} together, one can see that
$\bar{\ten{A}}_{i, i_2, \dots, i_m} \geq
\sqrt{\frac{2(m+1) p(1-q) \log n}{k^{m-1}}}
+ \frac{2(m+1)\log n}{3k^{m-1}}$
holds with probability less than $n^{-(m+1)}$.
Taking a union bound for $i_2,\dots, i_m$, we have
\[
\tnorminf{\mathcal{Q}_{\ten{Y}^\ast} (\lambda \ten{Z})} \leq (2m-1) \cdot \tnorminf{\bar{\ten{A}}}
\leq 3m\left(\sqrt{\frac{ p(1-q) \log n}{k^{m-1}}} + \frac{\log n}{k^{m-1}} \right)
\]
with probability at least $1-n^{-1}$.
Substitute the results above into \eqref{eq:suff_condition}, we obtain the sufficient condition
\begin{align}
p-q \geq C m(m-1) \sqrt{\frac{ p(1-q) mn \log m}{k^m}} + 6m \sqrt{\frac{ p(1-q) \log n}{k^{m-1}}} + 6m\frac{\log n}{k^{m-1}} \,.
\end{align}
Note that the order of the first two terms on the right-hand side are both dominated by $m^2 \sqrt{\frac{ p(1-q) mn \log m}{k^{m-1}}}$. Combining the first two terms leads to the sufficient condition of
$
p-q \geq C m^2 \sqrt{\frac{ p(1-q) mn \log m}{k^{m-1}}} + 6m \frac{\log n}{k^{m-1}} \,.
$
Then the first term on the right-hand side dominates the second term, if the mild condition
$(m^3 \log m)  p(1-q) k^{m-1} \geq 1$
is satisfied.
Then the remaining condition is
\begin{align}
\frac{p-q}{C\sqrt{ p(1-q) m^5 \log m}}
\geq
\sqrt{\frac{n}{k^{m-1}}} \,.
\end{align}
This completes our proof.
\end{proof}

redpajama_set_name: RedPajamaArXiv