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Since I wrote on the Poisson Binomial Distribution, I was bothered by a couple of things. The first and main problem that I had was with stack overflows and excessive $O\left( { n }^{ 2 } \right)$ memory allocation in method 2. As a reminder method 2 found in (Chen and Liu, 1997) looks like:

This may or may not look like gibberish to those who haven’t read the paper but I find Table 1 from the paper to be quite informative.

This diagram also drove me crazy until I had the epiphany that allowed me to rewrite method 2 non-recursively using only $O\left( n \right)$ space rather than $O\left( { n }^{ 2 } \right)$ auxiliary space. The epiphany was this.

## Epiphany

The diagram, when interpreted as a matrix, is a matrix in row echelon form. These zeros on the left are worthless to the algorithm and if we remove them and appropriately shift the remaining values to the left, the diagonal arrows in the diagram become vertical, downward arrows. This doesn’t affect the left-to-right arrows but this gives us a diagram that looks like a triangle. The fact that we have only downward and right arrows and a triangular shape helps a lot. It indicates that we can iterate over the rows then columns and use just a one-dimensional array, $r$, of size $n + 1$ to aggregate the computation because we only need values from the previous iteration of the outer loop and the previous value of the current inner loop to update the current value in the inner loop. Once the algorithm completes this array will contain the PMF. This means that we can use bottom-up dynamic programming to generate the PMF for the Poisson binomial distribution. At the end of each iteration of the inner loop, we add another value toward the end of the array that represents the probability of a certain number of successes, where the probability of zero successes is the last value in the array, the probability of one success is the second to last value, and so on. The only gotcha is that we need to normalize each value by multiplying by the normalizing constant from equation 5 in the paper:

Finally, we just have to reverse the array, $r$.

## Benefits

There are multiple benefits to writing the algorithm this way:

1. Stack overflows don’t occur.
2. Memory consumption is greatly reduced from $O\left( { n }^{ 2 } \right)$ to $O\left( n \right)$ and the $O\left( n \right)$ memory allocation is necessary to hold the PMF.
3. The actual speed of the algorithm is faster (wall clock, not asymptotic runtime).

## Remarks

This code doesn’t read like typical Scala code because it uses mutable arrays and while loops rather than functional constructs like folds. The purpose was speed and correctness over conciseness and elegance. This choice stems from the pmf algorithm’s runtime and space requirements: $O\left( { n }^{ 2 } \right)$ and $O\left( n \right)$, respectively.

The memory that is allocated contains the PMF are the end of the algorithm. Thus, no auxiliary space is necessary and the algorithm is optimally space efficient.

Another comment. The runtime for computing the full PMF is indeed $O\left( { n }^{ 2 } \right)$, but if we just want to know the probabilities for the first $k$ successes, the runtime is more like $O\left( k n \right)$. This is nice since the code was written to short circuit if the caller only requests the first $k$ probabilities. The code can also short circuit for using it as an inverse distribution function by specifying maxCumPr. The length - 1 of the resulting array is the IDF value.

## Tests

On my 2010 MacBook Pro, I can run val p = PoissonBinomial.pmf(Seq.fill(1000)(0.5)) in about 0.5 seconds and it’s accurate according to Wolfram Alpha. p(500) is 0.02522501817836080190684168876210234 and Wolfram Alpha says it’s 0.025225. p(1000) is 9.332636185032188789900895447238125E-302 and Wolfram Alpha says it’s 9.33264×10^-302.

What’s more important is that the algorithm based on BigDecimal can handle large probabilities whereas an implementation based on 64-bit IEEE 754 floats will eventually numerically overflow given a large enough set of probabilities. Avoiding this obviously comes at a cost. The cost is unfortunately both in terms of speed and memory. Because large probabilities make the normalizing constant small when there are a lot of probabilities in the computation, the BigDecimal values need to add a lot of precision and this is where the speed slowdowns and increased memory usage come in. So the runtime actually depends somewhat on the probability value inputs. This is OK because when the number of probabilities is small (< 1000), the runtime is reasonable (on the order of a few seconds). But when opting for an exact algorithm over an approximation, correctness should always trump speed. If this is not acceptable, thanks to the Central limit theorem, one can always use a (refined) normal approximation (see Poisson Binomial Distribution).


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Finding sums using Fourier

1. Apr 7, 2008

metalbec

This is a general question, I guess. If I am given an infinite series, how do I go about finding its sum using Fourier expansion?

2. Apr 7, 2008

quasar987

You find a function such that when expanded in a fourier series and then evaluated at a certain point coincide with the numerical series.

format: plain


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## Posts

Notes on a recent wave of spam 2018-06-14T15:39:51.090Z · score: 11 (7 votes)

Comment by rossry on When is pair-programming superior to regular programming? · 2019-10-10T13:09:51.406Z · score: 10 (6 votes) · LW · GW

It's easy to play armchair statistician and contribute little, but I want to point out that the empirics cited here are effectively just anecdotes. The paper studies 13 pairs and 13 individuals in three assignments in one class at UUtah. Its estimate of relative time costs is only significant to ~ because development time has variance of (if I backsolved correctly) 65%, which...seems about right. Still, it seems like borderline abuse of frequentist statistics to argue that a two-tailed p<0.05 should be required to reject the hypothesis that pairs finish projects in half the wall-clock time of individuals (which is the null the analysis assumes).

That said, the author correctly identifies that quality matters significantly more than speed. The quality metric, however, is "assignment tests passed" in throwaway academic projects, eliding the questions of what quality failures would or wouldn't be caught by the review / CI workflows that an industrial project would be going through anyway.

So, finger to the wind, this study feels like it suggests that a pair spends 15% more person-hours (once they get used to each other) before turning their schoolwork in, and do 15% more of the work of the assignment than a student working alone. Consistent with the higher reported work-enjoyment numbers! Definitely a stronger showing than I would have guessed! But definitely not well-abstracted by "no significant result for time; significant improvement for quality".

What am I missing here?

Comment by rossry on Eukryt Wrts Blg · 2019-09-29T01:59:11.495Z · score: 3 (3 votes) · LW · GW

(continued, to address a different point)

B and C seem like arguments against "simple" (i.e., even-odds) bets as well as weird (e.g., "70% probability") bets, except for C's "like bets where I'm surer...about what's going on", which is addressed by A (sibling comment).

Your point about differences in wealth causing different people to have different thresholds for meaningfulness is valid, though I've found that it matters much less than you'd expect in practice. It turns out that people making upwards of $100k/yr still do not feel good about opening up their wallet you give you$3. In fact, it feels so bad that if you do it more than a few times in a row, you really feel the need to examine your own calibration, which is exactly the success condition.

I've found that the small ritual of exchanging pieces of paper just carries significantly more weight than would be implied by their relation to my total savings. (For this, it's surprisingly important to exchange actual pieces of paper; electronic payments make the whole thing less real, ruining the whole point.)

Finally, it's hard to argue with someone's utility function, but I think that some rationalists get this one badly wrong by failing to actually multiply real numbers. For example, if you make a $10 bet (as defined in my sibling comment) every day for a year at the true probabilities, the standard decision of your profit/loss on the year is <$200, or $200/365 per day, which seems like a very small annual cost to practice being better calibrated and evaluate just how well-calibrated you are. Comment by rossry on Eukryt Wrts Blg · 2019-09-29T01:38:57.423Z · score: 3 (3 votes) · LW · GW Hi! I've done a fair amount of betting beliefs for fun and calibration over the years; I think most of these issues are solvable. A is a solved problem. The formulation that I (and my local social group) prefer goes like "The buyer pays$X*P% to the seller. The seller pays $X to the buyer if the event comes true." The precise payoffs aren't the important part, so long as they correspond to quoted probabilities in the correct way (and agreed sizes in a reasonable way). So this convention makes the probability you're discussing an explicit part of the bet terms, so people can discuss probabilities instead of confusing themselves with payoffs (and gives a clear upper bound for possible losses). Then you can work out exact payoffs later, after the bet resolves. (As a worked example, if you thought a probability was less than 70% and wanted to bet about$20 with me, if you "sold $20 at 70%" in the above convention, you'd either win$2070%=$14 or lose$20-($2070%)=$6. But it's even easier to see that you selling a liability of $20p(happens) for$2070% is good for you if you think p(happens)<70%.)

You've right that odds are a terrible convention for betting on probabilities unless you're trying to hide the actual numbers from your counterparties (which is the norm in retail sports betting).

Comment by rossry on Long-term Donation Bunching? · 2019-09-29T01:01:03.867Z · score: 1 (1 votes) · LW · GW

I also think that if the "sixth friend" donates $10k in line with each other friend's values and beliefs (as a result of social expectation, not contract), then there's no particular benefit to being the one who has to handle the money, and you don't need to trust in multi-year commitments. Comment by rossry on Long-term Donation Bunching? · 2019-09-29T00:53:06.260Z · score: 1 (1 votes) · LW · GW Your suggestion is correct, though it seemed too messy (and nonessential) to explain for the sake of an off-the-cuff proposal. I added a footnote to clarify this above, though. Comment by rossry on Long-term Donation Bunching? · 2019-09-28T00:41:56.340Z · score: 1 (1 votes) · LW · GW Proposal: Five friends in this situation write$10k checks[1] to a sixth. They all have a long chat about their altruist values and beliefs. The sixth donates $60k to a variety of EA causes. Question: Just how likely / unpleasant would the ensuing IRS audit be? (There's also a micro-donor-lottery version of this, except the individual contributions are personal gifts and the full$60k is a charitable donation.)

[1] Actually, you want this to be something like $7k, since the tax deduction from donating is worth [your marginal income tax rate] on the amount, roughly 30%. Formally,$10k less the tax benefits from donating $10k. Comment by rossry on Free Money at PredictIt? · 2019-09-27T07:42:56.215Z · score: 8 (4 votes) · LW · GW That's exactly correct. It's a standard taxation-begets-misallocation scenario. For reference, PI's current rules have this effect to roughly 0-3% per contract, potentially adding across multiple contracts in a bundle. Prices closer to 50% are worse (though prices further away have their own biases, as Zvi explains). Comment by rossry on Free Money at PredictIt? · 2019-09-27T07:32:02.184Z · score: 6 (4 votes) · LW · GW Yeah, Zvi is (unsurprisingly) right; the change in margining rules (after I wrote that post) makes it much better to sell the low-value contracts, and the withdrawal fees amortize if you're in for the longer term. To new rules, and on the back of my envelope, Zvi's 12% "arbitrage" is something like a few percent good: maybe it covers withdrawal fees on its own, and likely will do so after a few rounds. The opportunity cost of capital is a whole 'nother issue... I also strongly endorse the punchline that trading (even on the margins of trading costs) is some of the best rationalist training you can find. Comment by rossry on Free Money at PredictIt? · 2019-09-26T23:18:07.859Z · score: 1 (1 votes) · LW · GW Huh, I hadn't noticed that they didn't tie up the potential fees on your winnings. Hypotheses: • bug introduced when they moved from gross margining to net margining years and didn't reconsider fees withholding • doesn't actually matter; they don't give up ~anything by letting some people carrying small balances make free trades • it's really hard to abuse this into free trades repeatedly • the withholding here is too complicated and feel-bad to explain • other Comment by rossry on Taxing investment income is complicated · 2019-09-23T12:00:36.673Z · score: 3 (2 votes) · LW · GW Ah, that makes sense. Separately, I'm not entirely convinced by that second bullet point -- it seems like a non-omniscient state planner in a non-stationary environment would benefit from being able to determine the desired level of redistribution after the wealthy have accrued their income as wealth, rather than needing to get it right as they earned it. (I'm assuming away the confiscatory impulse here, naturally; in practice, the political economy of confiscation causes serious issues for deferred decisions about distribution like this.) Comment by rossry on Taxing investment income is complicated · 2019-09-22T23:40:55.720Z · score: 1 (1 votes) · LW · GW Can you explain more why the tax rate on the risk-free-rate portion of investment income should be 0? A positive rate here implements a proxy wealth tax (without raising the reporting problems of a direct wealth tax), and a nonzero wealth tax might be part of an optimal tax policy (e.g., for the lots-of-small-taxes argument, if no other reason). (I'm not sure that this is right, and am mostly asking this question from a stance of exploratory uncertainty.) Comment by rossry on Taxing investment income is complicated · 2019-09-22T23:35:36.191Z · score: 2 (2 votes) · LW · GW To be clear, this is the low-order approximation around 0; as explained in Paul's link (sibling to this) the effect away from zero involves the shape of the supply and demand curves through the relevant region of prices (and the stated claim holds when they're linear). Comment by rossry on Spiracular's Shortform Feed · 2019-09-12T22:53:59.965Z · score: 2 (2 votes) · LW · GW My guess is that one gets a reasonable start by framing more tasks as self-delegation (adding the middle step of "decide to do" / "describe task on to-do list" / "do task"), then periodically reviewing the tasks-competed list and pondering the benefits and feasibility of outsourcing a chunk of the "do task"s. Creating a record of task-intentions has a few benefits in making self-reflection possible; reflecting on delegation opportunities is a special case. Comment by rossry on Negative "eeny meeny miny moe" · 2019-08-21T14:55:24.967Z · score: 1 (1 votes) · LW · GW Oh, you're right. The net incentive to catch cheaters is actually... 1/(k(k-1)^2), then? The relative incentive story is worse, though still better in total than the positive version, and better still if you assume a constant-size disincentive to be caught cheating. Comment by rossry on Negative "eeny meeny miny moe" · 2019-08-20T10:49:46.809Z · score: 1 (1 votes) · LW · GW Another (related?) advantage is that the incentives to manipulate and catch manipulation are much better balanced with the negative ("you're out") version. Consider: • Perfectly cheating in the positive version improves your chances of winning by (n-1)/n, and stopping you from doing so improves each other person's chances by 1/n. • Perfectly cheating in a round of the negative version improves your chances of winning by 1/(k(k-1)), where k is the number of people in to start the round. Stopping you from doing so improves each other person's chances by the same amount. • The total (summed) incentive to manipulate in the negative version is (n-1)/n, the same as in the positive case. Comment by rossry on Open & Welcome Thread - August 2019 · 2019-08-13T13:17:23.744Z · score: 4 (6 votes) · LW · GW (Disclaimer: I'm a financial professional, but I'm not anyone's investment advisor, much less yours.) You mention diversity as an advantage because it reduces your risk, but this framing is missing the crucial point that you can transmute a portfolio that's 0.5x as risky as your baseline to one that returns twice as much as your baseline. (Wei_Dai mentions this, but obliquely.) The trick is to use leverage, which is not as hard to get (or as expensive, or as complicated) as you think. To be explicit about it: if you increase the per-dollar riskiness of your portfolio without increasing the per-dollar expected value, then after you leverage it down until it's at the optimal level of risk for your utility function (which you were going to do, right?), you will have lower expected returns. The relevant question is "how much lower?" (which is precisely to say "how much does it increase your per-dollar risk?"), which I answer in my response to Wei_Dai (nephew to this). The answer turns out to be "very little", but in order to get there, you have to be asking the right question first. Comment by rossry on Open & Welcome Thread - August 2019 · 2019-08-13T13:13:58.042Z · score: 4 (2 votes) · LW · GW I don't have a good intuition about how costly this actually is in practice, if you only play with 10% of your portfolio. tl;dr extremely little. Here's some numbers I made up: • Let the market's single common factor explain 90% of the variance of each stock. • Let the remaining 10%s be idiosyncratic and independent. • Let stocks have equal volatility (and let all risk be described by volatility). Now compare a portfolio that's$100 of each of a hundred stocks with one that's $90 of each of a hundred plus$1k of another stock. (I'll model each stock as 0.75 times the market factor plus 0.25 a same-variance idiosyncratic factor.) Compared to a $10k single-stock portfolio... • the equal-weighted portfolio has like • the shot-caller's portfolio has like ...for an increase in of 4.5 basis points. So, pretty negligible. Even if the market's single factor explains only half of the variance of each stock, the increased risk of the shot-caller's portfolio is just 40 basis points ( vs ). In the extreme case where stocks are uncorrelated, the increased risk is +34.5%, though I think that that's unrealistically generous to the diversification strategy. Since an increase in volatility-per-dollar of basis points means that you give up basis points of your expected returns, I'm going to say that this effect is negligible in the "10% of portfolio" setting. Comment by rossry on Does human choice have to be transitive in order to be rational/consistent? · 2019-08-11T05:29:46.881Z · score: 1 (1 votes) · LW · GW Can you define "rational/consistent"? The terms are a bit overloaded, especially in this community, and making your definition precise is itself most of the answer to your question. For example, you give some good examples of nontransitive decider-actors, and if some of them are "rational", then nontransitive preferences can be rational, as you point out. Alternatively, one definition of "consistent" is that a decider-actor will always reach the same decision when it has the same information, regardless of what other options it has previously rejected, which requires transitive preferences. Comment by rossry on How can we respond to info-cascades? [Info-cascade series] · 2019-08-05T11:08:29.548Z · score: 1 (1 votes) · LW · GW Is the idea something like: the actor themself is uncertain about the value of the project, and the kickstarter also helps them find out whether it's worth doing Nope! The paper's model for this result assumes that the value conditioned on success is known to the proposer, so that the proposer's only incentive is to maximize their own profits by setting the payout odds and threshold. The (non-obvious to me) result that the paper proves is that this coincidentally minimizes the probability of up-cascades: A higher decision threshold excludes more DOWN cascades while it is less likely to be reached. We show that the concern about potential DOWN cascades dominates the concern about likelihood to reach the target. To maximize the proceeds, the proponent endogenously sets the target to the smallest number that in equilibrium completely excludes DOWN cascades in the same spirit as Welch (1992), with the caveat that the proponent utilizes both price and target to achieve this. Consequently, with endogenous issuance pricing, there is no DOWN cascade which stops private information aggregation, and good projects are always financed while bad projects are never financed, when the crowd base N becomes very large. In other words, financing efficiency and information aggregation efficiency approaches the first best as N grows bigger, despite the presence of information cascades. Comment by rossry on Zeno walks into a bar · 2019-08-04T09:12:12.500Z · score: 10 (6 votes) · LW · GW My (uninformed) guess is "humor, of the slightly mind-sharpening sort". Comment by rossry on Proposed algorithm to fight anchoring bias · 2019-08-03T06:41:02.812Z · score: 11 (5 votes) · LW · GW Hypothesis: (formed before running the test below) "clearly too high" will be more of an overestimate than "clearly too low" is an underestimate; most midpoints will be high; this will bias estimates high (especially through repeated self-anchoring to high midpoints). Anecdata: I tried using this method to estimate the number of gates in Hong Kong International Airport. (I'm currently standing in a line in HKIA, and have flown through it at least a dozen times in my life.) • 40 ~ 10,000 • 40 ~ 5,020 • 40 ~ 2,530 • 40 ~ 1285 • 40 ~ 662 (maybe 351 is neither obviously too high or too low, so I should take it? or maybe it's slightly high and so...) • 40 ~ 351 (195 I don't have a strong opinion about, so I'll take it) So, depending on how I decide to stop, I get estimates around 195 or 351. Apparently the true answer is 90, supporting my hypothesis. Proposal: Maybe use geometric mean instead of arithmetic mean? I tried to clear my head and try again with geomean: • 40 ~ 10,000 • 40 ~ 632 (ehhh, 159 is more likely to high than too low) • 40 ~ 159 (80 is too low) • 80 ~ 159 (113) There's one suspect step around 159, but my guess here is that being anchored by one higher number (632) rather than five (5020, 2530, 1285, 662, 351) is enough to actually make me think "probably less than 159" rather than "no strong opinion about 195". Terminating at 159 also is an outperformance of arithmetic mean, but feels a bit more like luck around where the midpoints fell. Comment by rossry on Get Rich Real Slowly · 2019-06-12T08:05:54.069Z · score: 3 (2 votes) · LW · GW What's up with "FDIC insurance up to$1,000,000"? Wikipedia claims that FDIC insurance only covers 0.25mln/bank unless I have a joint account, retirement account, or some kind of legal entity which I'm not. Comment by rossry on Fractional Reserve Charity · 2019-06-07T12:46:58.638Z · score: 4 (2 votes) · LW · GW I don't think that there is off-the-shelf insurance for "I lose my job (prospects) or otherwise choose to have lower lifetime earnings, in some way I did not foresee." It would be confusing to me if most EAs had equity investments they could not afford to bear a crash loss in, especially those involved in an emergency fund scheme. Why add market exposure to the portfolio of donations + scheme membership + liquid cash? (Especially, why as market exposure you expect other scheme participants to share?) That said, it is possible to buy insurance against a market crash. Probably not as a centralized service. Comment by rossry on 0.999...=1: Another Rationality Litmus Test · 2019-05-26T22:41:10.149Z · score: 3 (2 votes) · LW · GW So shouldn't that inequality apply to 0.AAA... (base eleven) and 0.999... (base ten) as well? (A debatable point maybe). Not debatable, just false. Formally, the fact that for all does not imply that . If I were to poke a hole in the (proposed) argument that 0.[k 9s]{base 10} < 0.[k As]{base 11} (0.9<0.A; 0.99<0.AA;...), I'd point out that 0.[2*k 9s]{base 10} > 0.[k As]{base 11} (0.99>0.A; 0.9999>0.AA;...), and that this gives the opposite result when you take (in the standard sense of those terms). I won't demonstrate it rigorously here, but the faulty link here (under the standard meanings of real numbers and infinities) is that carrying the inequality through the limit just doesn't create a necessarily-true statement. 0.111...{binary} is 1, basically for the Dedekind cut reason in the OP, which is not base-dependent (or representation-dependent at all) -- you can define and identify real numbers without using Arabic numerals or place value at all, and if you do that, then 0.999...=1 is as clear as not(not(true))=true. Comment by rossry on No Really, Why Aren't Rationalists Winning? · 2019-05-22T23:09:55.839Z · score: 6 (2 votes) · LW · GW I think your comment is unnecessarily hedged -- do you think that you'd find much disagreement among LWers who interact with FHI/GMU-Econ over whether people there sometimes (vs never) fail to do level-one things? I think I understand the connotation of your statement, but it'd be easier to understand if you strengthened "sometimes" to a stronger statement about academia's inadequacy. Certainly the rationality community also sometimes fails to do even the obvious, level-one intelligent character things to enable them to achieve their goals -- what is the actual claim that distinguishes the communities? Comment by rossry on No Really, Why Aren't Rationalists Winning? · 2019-05-22T12:30:33.702Z · score: 3 (2 votes) · LW · GW I'm confused; it seems like evidence against the claim that you can get arbitrary amounts of value out of learning generic rationality skills, but I don't see it as "devastating" to the claim you can get significant value, unless you're claiming that "spent years learning all that stuff, and now do it as a day job; some of them 16 hours a day" should imply only a less-than-significant improvement. Or am I missing something here? Comment by rossry on Counterspells · 2019-04-29T14:30:12.832Z · score: 3 (2 votes) · LW · GW cf. my comment cousin to this one; I misunderstood what the term was pointing at at first, though I stand by my complaint that that's a problem with the term. Comment by rossry on Counterspells · 2019-04-28T14:34:51.922Z · score: 3 (3 votes) · LW · GW Thanks; I legitimately misunderstood at first read whether "counterspell" was intended to apply to the invocations thrown out by bad arguers or the concise and specific distillations the OP is presenting for use. On re-read, I agree that it's supposed to be a set of useful tools. I remain convinced of the specific claim that "counterspell" is bad jargon (though I don't think it's good practice to cite my own confusion too strongly; the incentives there aren't great). I agree that MtG''s paradigm where more general counterspells are more expensive seems like a good fit for thinking about rhetorical (and perhaps epistemic) tactics, though I reiterate that that's not how they work in many other settings, and that ambiguous baggage is worse than no baggage for this sort of thing. The question of whether identifying counterspells with magic is supposed to be a positive or negative association is additional gratuitous confusion -- I think your claim that the magic metaphor implies they don't work is wrong, but I'm not 85% sure. Comment by rossry on Counterspells · 2019-04-28T04:40:42.545Z · score: 16 (9 votes) · LW · GW Strong approval of the overall goal of the post, but here's a semantic criticism: In accordance with the Rationalist tradition that requires everything to have a nerdy sci-fi or fantasy name I parse this as an in-joke (and appreciate it as such), but I do think that regularly minting new jargon that's likely to carry substantial, conflicting(!) contextual baggage (not all of it appropriate[1]) is...a bad norm for an epistemic community to have. I also think that the deeper tradition of jargon-forging (as Eliezer practiced it) involved names that sounded nerdy, but _not_ sci-fi or fantasy -- _cf._ Knowing About Biases Can Hurt People, which uses "Fully General Counterargument" in much the same way(?) as you're using "Counterspell". "Fully General Counterargument" is slightly more unwieldy, but apart from that is a better piece of jargon -- it's less loaded and by leaning even harder into the implicit snark that winks at the fact that the purported counter- isn't working at all, makes it even more clear that no, this is not a useful thing. [1] To belabor this point, MtG counterspells are fully general (edit: okay, not **fully** general, and see Slider below), and a reasonable fit for the term as you're using it, but D&D (3.5e, at least) counterspells are based on negating a spell by casting a copy of it, which is not what you mean at all. I don't actually know what "counterspells" are in WoD/Mage, but the risk that they're even further afield from your intention should be another strike against using the already-loaded handle. Comment by rossry on Open Thread April 2019 · 2019-04-28T04:23:18.363Z · score: 1 (1 votes) · LW · GW preference "my decisions should be mine" - and many people seems to have it Fair. I'm not sure how to formalize this, though -- to my intuition it seems confused in roughly the same way that the concept of free will is confused. Do you have a way to formalize what this means? (In the absence of a compelling deconfusion of what this goal means, I'd be wary of hacking epistemics in defense of it.) There are "friends" who claim to have the same goals as me, but later turns out that they have hidden motives. Agreed and agreed that there's a benefit to removing their affordance to exploit you. That said, why does this deserve more attention than the inverse case (there are parties you do not trust who later turn out to have benign motives)? Comment by rossry on Open Thread April 2019 · 2019-04-27T02:57:39.037Z · score: 3 (2 votes) · LW · GW "give the power over my final decisions to small random events around me" seems like a slightly confused concept if your preferences are truly indifferent. Can you say more about why you see that as a problem? The potential adversary seems like a more straightforward problem, though one exciting possibility is that lightness of decisions lets a potential cooperator manipulate your decisions on favor of your common interests. And presumably you already have some system for filtering acquaintances into adversaries and cooperators. Is the concern that your filtering is faulty, or something else? [Commitment] eventually often turn to be winning strategy, compared to the flexible strategy of constant updating expected utility. Some real-world games are reducible to the game of Chicken. Commitment is often a winning strategy in them. Though I'm not certain that it's a commitment to a particular set of beliefs about utility so much as a more-complex decision theory which sits between utility beliefs and actions. In summary, if the acquaintances whose info you update on are sufficiently unaligned with you and your decision theory always selects the addition that your posterior assigns the highest utility, then your actions will be "over-updating on the evidence" if your beliefs are properly Bayesian. But I don't think the best response is to bias yourself towards under-updating. Comment by rossry on Moral Weight Doesn't Accrue Linearly · 2019-04-23T23:22:08.456Z · score: 3 (2 votes) · LW · GW Do you bite the bullet that this means the set of things you morally value changes discontinuously and predictably as things move out of your light cone? (Or is there some way you value things less as they are less "in" your light cone, in some nonbinary way?) Comment by rossry on On the Nature of Programming Languages · 2019-04-23T23:15:02.894Z · score: 4 (3 votes) · LW · GW I think it's from SICP that programs are meant to be read by humans and only incidentally for computers to execute; I've been trying for more than a year now to write a blog post about the fundamental premise that, effort-weighted, we almost never write new programs from scratch, and mostly are engaged in transmuting one working program into another working program. Programs are not only meant to be read by humans, but edited by humans. I think if you start from the question of how much effort it is to write a new program on a blank page, most languages will come out looking the same, and the differences will look like psychological constructs. If you ask, however, how much effort it is to change an existing piece of a code base to a specific something else, you start to see differences in epistemic structure, where it matters how many of the possible mutations that a human algorithm might try will non-obviously make the resulting program do something unexpected. And that, as you point out, opens the door to at least some notion of universality. Comment by rossry on On the Nature of Programming Languages · 2019-04-23T22:55:11.028Z · score: 3 (2 votes) · LW · GW C, the most widespread general-purpose programming language, does things that are extremely difficult or impossible in highly abstract languages like Haskell or LISP Can you give an example? I'm surprised by this claim, but I only have deep familiarity with C of these three. (My primary functional language includes mutable constructs; I don't know how purely functional languages fare without them.) Comment by rossry on Open Thread April 2019 · 2019-04-09T23:40:27.223Z · score: 1 (1 votes) · LW · GW I was trying to construct a proof along similar lines, so thank you for beating me to it! Note that 2 is actually a case of 1, since you can think of the "walls" of the simplex as being bets that the universe offers you (at zero odds). Comment by rossry on Open Thread April 2019 · 2019-04-09T11:34:57.791Z · score: 1 (1 votes) · LW · GW (This comment isn't an answer to your question.) If I'm understanding properly, you're trying to use the set of bets offered as evidence to infer the common beliefs of the market that's offering them. Yet from a Bayesian perspective, it seems like you're assigning P( X offers bet B | bet B has positive expectation ) = 0. While that's literally the statement of the Efficient Markets Hypothesis, presumably you -- as a Bayesian -- don't actually believe the probability to be literally 0. Getting this right and generalizing a bit (presumably, you think that P( X offers B | B has expectation +epsilon ) < P( X offers B | B has expectation +BIG_E )), should make the market evidence more informative (and cases of arbitrage less divide-by-zero, break-your-math confusing). Comment by rossry on Open Thread April 2019 · 2019-04-09T09:46:50.312Z · score: 1 (1 votes) · LW · GW I'm confused what the word "fairly" means in this sentence. Do you mean that they make a zero-expected-value bet, e.g., 1:1 odds for a fair coin? (Then "fairly" is too strong; non-degenerate odds (i.e., not zero on either side) is the actual required condition.) Do you mean that they bet without fraud, such that one will get a positive payout in one outcome and the other will in the other? (Then I think "fairly" is redundant, because I would say they haven't actually bet on the outcome of the coin if the payouts don't correspond to coin outcomes.) Comment by rossry on How can we respond to info-cascades? [Info-cascade series] · 2019-03-17T01:53:12.102Z · score: 3 (2 votes) · LW · GW A related idea in non-punishment of "wrong" reports that have insufficient support (again in the common-prior/private-info setting) comes from this paper [pdf] (presented at the same conference), which suggests collecting reports from all agents and assigning rewards/punishments by assuming that agents' reports represent their private signal, computing their posterior, and scoring this assumed posterior. Under the model assumptions, this makes it an optimal strategy for agents to truly reveal their private signal to the mechanism, while allowing the mechanism to collect non-cascaded base data to make a decision. In general, I feel like the academic literature on market design / mechanism design has a lot to say about questions of this flavor. Comment by rossry on How can we respond to info-cascades? [Info-cascade series] · 2019-03-17T01:34:53.337Z · score: 10 (3 votes) · LW · GW Abstract: Considering information cascades (both upwards and downwards) as a problem of incentives, better incentive design holds some promise. This academic paper suggests a model in which making truth-finding rewards contingent on reaching a certain number of votes prevents down-cascades, and where an informed (self-interested) choice of payout odds and threshold can also prevent up-cascades in the limit of a large population of predictors. 1) cf. avturchin from the question about distribution across fields, pointing out that up-cascades and down-cascades are both relevant concerns, in many contexts. 2) Consider information cascades as related to a problem of incentives -- in the comments of the Johnichols post referenced in the formalization question, multiple commentators point out that the model fails if agents seek to express their marginal opinion, rather than their true (posterior) belief. But incentives to be right do need to be built into a system that you're trying to pump energy into, so the question remains of whether a different incentive structure could do better, while still encouraging truth-finding. 3) Up-Cascaded Wisdom of the Crowd (Cong and Xiao, working paper) considers the information-aggregation problem in terms of incentives, and consider the incentives at play in an all-or-nothing crowdfunding model, like venture capital or Kickstarter (assuming that a 'no' vote is irrevocable like a 'yes' vote is) -- 'yes' voters win if there is a critical mass of other 'yes' voters and the proposition resolves to 'yes'; they lose if there is a critical mass and the proposition resolves to 'no'; they have 0 loss/gain if 'yes' doesn't reach a critical mass; 'no' voters are merely abstaining from voting 'yes'. Their main result is that if the payment of incentives is conditioned on the proposition gaining a fixed number of 'yes' votes, a population of symmetric, common-prior/private-info agents will avoid down-cascades, as a single 'yes' vote that breaks a down-cascade will not be penalized for being wrong unless some later agent intentionally votes 'yes' to put the vote over the 'yes' threshold. (An agent i with negative private info still should vote no, because if a later agent i' puts the vote over the 'yes' threshold based in part on i's false vote, then i expects to lose on the truth-evaluation, since they've backed 'yes' but believe 'no'.) A further result from the same paper is that if the actor posing the proposition can set the payout odds and the threshold in response to the common prior and known info-distribution, then a proposition-poser attempting to minimize down-cascades (perhaps because they will cast the first 'yes' vote, and so can only hope to win if the vote resolves to 'yes') will be incentivized to set odds and a threshold that coincidentally minimize the chance of up-cascades. In the large-population limit, the number of cascades under such an incentive design goes to 0. 4) I suspect (but will not here prove) that augmenting Cong and Xiao's all-or-nothing "crowdfunding for 'yes'" design with a parallel "crowdfunding for 'no'" design -- i.e., 'no' voters win (resp. lose) iff there is a critical mass of 'no' voters and the proposition resolves 'no' (resp. 'yes') -- can further strengthen the defenses against up-cascades (by making it possible to cast a more informed 'no' vote conditioned on a later, more-informed agent deciding to put 'no' over the threshold). Comment by rossry on How can we respond to info-cascades? [Info-cascade series] · 2019-03-17T01:30:21.056Z · score: 0 (0 votes) · LW · GW [this answer was duplicated when I mistakenly copied my comment into an answer and then moved the comment to an answer.] Comment by rossry on Distribution of info-cascades across fields? [Info-cascade series] · 2019-03-17T00:14:15.158Z · score: 1 (1 votes) · LW · GW Why stop at 2? Belief-space is large, and many issues admit more than one (+/-) bit of information to cascade. Comment by rossry on Open Thread February 2019 · 2019-03-02T02:03:26.197Z · score: 1 (1 votes) · LW · GW But how make small changes in the trajectory of a star? One idea is to impact the star with large comets. It is not difficult, as remote Oort cloud objects (or wandering small planets, as they are not part of already established orbital movement of the star) need only small perturbations to start falling down on the central star, which could be done via nuclear explosions or smaller impacts by smaller astreoids. I don't think this works; conservation of momentum means that the impact is almost fully counteracted by the gravitational pull that accelerated the comet to such speed (so that, in the end, the delta-v imparted to the star is precisely what you imparted with your nuclear explosions or smaller asteroids). Maybe a Shkadov thruster is what you want? (It's slow going, though; this article suggests 60ly/200My, accounting for acceleration.) Comment by rossry on Blackmail · 2019-02-23T10:35:22.371Z · score: 1 (1 votes) · LW · GW Winner's Curse doesn't seem like the right effect to me -- it seems more like an orthogonality/Goodhart effect, where optimizing for outrageousness decreases the fitness w/r/t social welfare (on the margin). It's always in the blackmailer's interest to make the outrageousness greater, so they're not (selfishly) sad when they overshoot. Comment by rossry on Can We Place Trust in Post-AGI Forecasting Evaluations? · 2019-02-19T02:11:42.887Z · score: 8 (4 votes) · LW · GW I'm still thinking about what quantitative estimates I'd stand behind. I think I'd believe that a prize-based competitive prediction system with all eval deferred until and conditioned on AGI is <4% to add more than1mln of value to [just pay some smart participants for their best-efforts opinions].

(If I thought harder about corner-cases, I think I could come up with a stronger statement.)

Comment by rossry on Can We Place Trust in Post-AGI Forecasting Evaluations? · 2019-02-18T23:39:13.713Z · score: 3 (3 votes) · LW · GW

If the reason your questions won't resolve is that you are dead or that none of your money at all will be useful, I think things are a bit different.

I'm confused; to restate the above, I think that a p% chance that your predictions don't matter (for any reason: game rained out, you're dead, your money isn't useful) is (to first order) equivalent to a p% tax on investment in making better predictions. What do you think is different?

one major ask is that the forecasters believe the AGI will happen in between, which seems to me like an even bigger issue

Sure, that's an issue, but I think that requiring participants to all assume short AGI timelines is tractable in a way that the delayed/improbable resolution issues are not.

I can imagine a market without resolution issues that assumes participants all believe short AGI timelines could support 12 semi-professional traders subsidized by interested stakeholders. I don't believe that a market with resolution issues as above can elicit serious investment in getting its answers right from half that many. (I recognize that I'm eliding my definition of "serious investment" here.)

Comment by rossry on Can We Place Trust in Post-AGI Forecasting Evaluations? · 2019-02-18T14:39:01.178Z · score: 3 (3 votes) · LW · GW

I understand Zvi's points as being relatively universal to systems where you want to use rewards to incentivize participants to work hard to get good answers.

No matter how the payouts work, a p% chance that your questions don't resolve is (to first order) equivalent to a p% tax on investment in making better predictions, and a years-long tie-up kills iterative growth and selection/amplification cycles as well limiting the return-on-investment-in-general-prediction-skill to a one-shot game. I don't think these issues go away if you reward predictions differently, since they're general features of the relation between the up-front investment in making better predictions and the to-come potential reward for doing so well.

(A counterpoint I'll entertain is Zvi's caveat to "quick resolution" -- which also caveats "probable resolution" -- that sufficient liquidity can substitute for resolution. But bootstrapping that liquidity itself seems like a Hard Problem, so I'd need to further be convinced that it's tractable here.)

Comment by rossry on Can We Place Trust in Post-AGI Forecasting Evaluations? · 2019-02-18T13:15:17.761Z · score: 2 (4 votes) · LW · GW

Assuming that your AI timelines are well-approximated by "likely more than three years", Zvi's post on prediction market desiderata suggests that post-AGI evaluation is pretty dead-on-arrival for creating liquid prediction markets. Even laying aside the conditional-on-AGI dimension, the failures of "quick resolution" (years) and "probable resolution" (~20%, by your numbers) are crippling for the prospect of professionals or experts investing serious resources in making profitable predictions.

Comment by rossry on Probability space has 2 metrics · 2019-02-11T06:23:16.236Z · score: 8 (4 votes) · LW · GW

The speculative proposition that humans might only be using one metric rings true and is compellingly presented.

However, I feel a bit clickbaited by the title, which (to me) implies that probability-space has only two metrics (which isn't true, as the later proposition depends on). Maybe consider changing it to "Probability space has multiple metrics", to avoid confusion?

Comment by rossry on Open Thread February 2019 · 2019-02-08T13:23:33.873Z · score: 9 (5 votes) · LW · GW

It varies a lot between papers (in my experience) and between fields (I imagine), but several hours for a deep reading doesn't seem out of line. To take an anecdote, I was recently re-reading a paper (of comparable length, though in economics) that I'm planning to present as a guest of a mathematics reading group, and I probably spent 4 hours on the re-read, before starting on my slides and presumably re-re-reading a bunch more.

Grazing over several days (and/or multiple separate readings) is also my usual practice for a close read, fwiw.

Comment by rossry on X-risks are a tragedies of the commons · 2019-02-07T09:29:36.884Z · score: 2 (2 votes) · LW · GW

I acknowledge that there's a distinction, but I fail to see how it's important. When you (shut up and) multiply out the probabilities, the expected personal reward for putting in disproportionate resources is negative, and the personal-welfare-optimizing level of effort is lower than the social-welfare-optimizing level.

Why is it important that that negative expected reward is made up of a positive term plus a large negative term (X-risk defense), instead of a negative term plus a different negative term (overgrazing)?


text:
# On using \cite or something else to group successive references together [duplicate]

Possible Duplicate:
Citing a range of papers (using numeric keys)?

Quick question. Let's say I have five references to cite which are A to E. Currently, I'm using \cite{A,B,C,D,E} which produces

[1,2,3,8,11]

How do I modify it so that it produces the output of

[1-3,8,11]

or at least it groups the consecutive references together so as not to have a very long output format? I hope this is possible.

-

## marked as duplicate by lockstep, Stefan Kottwitz♦Nov 20 '12 at 15:55

Welcome to TeX.sx! This question is very similar to tex.stackexchange.com/questions/3871/…. Please take a look at it as the information there might help you. If so, that's great, and we'll probably close this question as a duplicate just to keep the place tidy and to help people find answers quickly. If not, please edit your question here to explain why so that people can better focus their attention to help you. –  lockstep Nov 19 '12 at 19:04
Thanks for the link. I tried \usepackage[numbers,sort&compress]{natbib} as suggested in the link and continue to use \cite{A,B,C,D,E} in the body. However, it did not do the job. I wonder if I missed out something. –  Ivan Nov 19 '12 at 19:22
Without a minimal working example that shows the problem, it's hard to say. –  lockstep Nov 19 '12 at 20:07
Without loading natbib you can try with \usepackage{cite} –  egreg Nov 20 '12 at 0:10
This is what I had in my latex file --> "....balance data rate and complexity \cite{Lee,Li,Manchon}" and it comes out "....balance data rate and complexity [27, 28, 29]" although I declared the natbib package with the preambles as above. @egreg, must I include any preambles for the cite package? Thanks for you help. Appreciate it. –  Ivan Nov 20 '12 at 14:34

Ok I have found the complete solution to my question. I will share it here just in case it is useful to others as most sites do not provide the answer explicitly. Special thanks to @lockstep and @egreg for providing helpful comments.

I will use the cite package as the solution. Just include the following statements when you declare your latex packages.

\usepackage{cite} \def\citepunct{], [} \def\citedash{]--[}

To utilize it, simply use \cite{} for your references. Taking the alphabets to mean numerical progression, we have the following examples:

\cite{RefA, RefB, RefC, RefD} --> [1]-[4]

while,

\cite{RefA, RefC, RefD, RefE, RefG} --> [1],[3]-[5],[7]

Hope this helps.

-


text:
1. ## Finding the domain

The question:
Determine f/g, and give the domain.

$\displaystyle f(x) = \sqrt{x -1}$
$\displaystyle g(x) = x - \sqrt{x + 1}$

I did:

$\displaystyle \frac{\sqrt{x-1}}{x - \sqrt{x + 1}}$

And I found that x had to be >= 1 (since you can't square root a negative value). I wrote [1, inf) as my answer. However, the answer is $\displaystyle [1, \frac{1 + \sqrt{5}}{2}) U (\frac{1 + \sqrt{5}}{2}, \inf)$

I'm not sure how this answer was calculated. Any assistance would be awesome. Thanks.

2. x-sqrt(x+1)!=0

3. Aha! That was a massive oversight.

For other people looking at this thread:

$\displaystyle x - \sqrt{x + 1} = 0$
$\displaystyle x + 1 = x^2$
$\displaystyle x^2 -x - 1 = 0$

$\displaystyle \frac{1 +- \sqrt{1 - 4(1)(-1)}}{2}$
$\displaystyle \frac{1 +- \sqrt{5}}{2}$

Erm, how do I do "+ or -" in Latex so this quadratic eqn makes more sense?

4. but solution with minus is not induced(why?)

5. Because we already know that x must be larger than or equal to 1 to satisfy the numerator.

6. Job well done!

"Erm, how do I do "+ or -" in Latex so this quadratic eqn makes more sense?"

\pm

$\displaystyle \pm$


text:
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Imaging soliton dynamics in optical microcavities

Abstract

Solitons are self-sustained wavepackets that occur in many physical systems. Their recent demonstration in optical microresonators has provided a new platform for the study of nonlinear optical physics with practical implications for miniaturization of time standards, spectroscopy tools, and frequency metrology systems. However, despite its importance to the understanding of soliton physics, as well as development of new applications, imaging the rich dynamical behavior of solitons in microcavities has not been possible. These phenomena require a difficult combination of high-temporal-resolution and long-record-length in order to capture the evolving trajectories of closely spaced microcavity solitons. Here, an imaging method is demonstrated that visualizes soliton motion with sub-picosecond resolution over arbitrary time spans. A wide range of complex soliton transient behavior are characterized in the temporal or spectral domain, including soliton formation, collisions, spectral breathing, and soliton decay. This method can serve as a visualization tool for developing new soliton applications and understanding complex soliton physics in microcavities.

Introduction

Temporal solitons are indispensable in optical fiber systems1 and exhibit remarkable nonlinear phenomena2. The potential application of solitons to buffers and memories3,4, as well as interest in soliton physics has stimulated approaches for experimental visualization of multi-soliton trajectories. Along these lines, the display of soliton trajectories in a co-moving frame5 allows an observer to move with the solitons and is being used to monitor soliton control and interactions of all types in fiber systems5,6,7,8,9,10,11,12. However, this useful data visualization method relies upon soliton pulse measurements that are either limited in bandwidth (pulse resolution) or record length. It is therefore challenging to temporally resolve solitons over the periods often required to observe their complete evolution. For example, the time-lens method13 can provide the required femtosecond-resolution, but has a limited record length set by the pump pulse. Also, while the relative position of closely spaced soliton complexes11 can be inferred over time from their composite dispersive Fourier transform (DFT) spectra14, Fourier inversion requires the constituent solitons to have similar waveforms which restricts the generality of the technique. Efforts that combine these two methods were also reported very recently15,16.

These limitations are placed in sharp focus by recent demonstrations of soliton generation in microcavities17,18,19,20,21,22,23. This new type of dissipative soliton24 was long considered a theoretical possibility3 and was first observed in optical fiber resonators4. Their microcavity embodiment poses severe challenges for imaging of dynamical phenomena by conventional methods, because multi-soliton states feature inherently closely spaced solitons. Preliminary real-time measurements using time lens25, and direct detection26 have been explored, but were limited in either recording length or pulse resolution. Nonetheless, the compactness of microcavity-based soliton systems has practical importance for miniaturization of frequency comb technology27 through chip-based microcombs28,29. Indeed, spectroscopy systems30,31, coherent communication32, ranging33,34, and frequency synthesis35 demonstrations using the new miniature platform have already been reported. Moreover, the unique physics of the new soliton microcavity system has led to observation of many unforeseen physical phenomena involving compound soliton states, such as Stokes solitons36, soliton number switching37 and soliton crystals38.

In this work, we report imaging of a wide range of soliton phenomena in microcavities. Soliton formation, collisions10, breathing6,39,40,41, Raman shifting42,43, as well as soliton decay are observed. Significantly, femtosecond-time-scale resolution over arbitrary time spans (distances) is demonstrated (and required) in these measurements. Also, real-time spectrograms are produced along-side high-resolution soliton trajectories. These features are derived by adapting coherent linear optical sampling44 and electric-field cross-correlation45 to the problem of microcavity soliton imaging. Beyond the necessity to employ a new method for imaging soliton motion in microcavities, the high-repetition rate of microcavity solitons (tens of gigahertz and higher) is advantageous in sampling-based recording of motion.

Results

Coherent sampling of soliton motion

To image the soliton trajectories, a separate optical probe pulse stream is generated at a pulse rate that is close to the rate of the solitons to be imaged in the microcavity. The small difference in these rates causes a pulse-to-pulse temporal shift of the probe pulses relative to the microcavity signal pulses as illustrated in Fig. 1a. By heterodyne detection of the combined streams, the probe pulses coherently sample the microcavity signal producing a temporal interferogram46,47 shown in Fig. 1a. Ultimately, the time shift per pulse accumulates so that the sampling repeats in the interferogram at the frame rate which is described below, and is close in value to the difference of sampling and signal rates. Probe pulses have a sub-picosecond temporal resolution that enables precise monitoring of the temporal location of the soliton pulses. Moreover, the coherent mixing of probe and soliton pulses allows extraction of each soliton’s spectral evolution by fast Fourier transform of the interferogram. In principle, the probe pulses can be generated by a second microcavity soliton source operating in steady state. However, in the present measurement, an electro-optic (EO) comb is used45,48,49. The EO comb pulse rate is conveniently adjusted electronically to match the rates of various phenomena being probed within the microcavity.

The soliton signal is produced by a 3 mm diameter silica wedge resonator with a free-spectral-range (FSR) of 22 GHz and intrinsic quality factor above 200 million18,50. The device generates femtosecond soliton pulses when pumped at frequencies slightly lower than a cavity resonant frequency18. To sample the 22 GHz soliton signal the EO comb was formed by modulation of a tunable continuous-wave (CW) laser. The EO comb features ~1.3 THz optical bandwidth (within 1 dB power variation) and an 800 fs full-width-at-half-maximum (FWHM) pulse width is measured by frequency-resolved optical gating (FROG) and autocorrelation as shown in Fig. 1b. Further details on the experimental setup are provided in the Methods section. In all presented measurements, the pump laser of the resonator scans linearly from higher to lower frequency to initiate parametric oscillation51 in the microcavity followed by chaotic dynamics. Ultimately, step-like features are observable in the resonator transmitted power (Fig. 1c) indicating the formation of soliton states17. The typical pump power and laser scan speed are ~70 mW and ~1 MHz/μs, respectively.

Measuring multiple soliton trajectories

As described above, heterodyne detection of the soliton signal and the EO-comb pulse produces the electrical interferogram. The period of the signals in the interogram is adjusted by tuning the EO-comb repetition rate. In the initial measurements, it is set to ~10 MHz lower than the rate of the microcavity signal so that the nominal period in the interferogram is ~100 ns. To display the interferogram signal a co-rotating frame is applied. First, a frame period T is chosen that is close to the period of signals of interest in the interferogram. Integer steps (i.e., mT) are plotted along the x-axis while the interferogram is plotted along the y-axis, but offset in time by the x-axis time step (i.e., t-mT). To make connection to the physical time scale of the solitons, the y-axis time scale is compressed by the same bandwidth compression factor (T×FSR) that accompanies the sampling process (see Discussion). The y-axis scale is accordingly set to span one microcavity round-trip time. A typical measurement plotted in this manner is given in Fig. 1d. Because this way of plotting the data creates a co-rotating reference frame, a hypothetical soliton pulse with an interferogram period equal to the frame rate T would appear as a horizontal line in Fig. 1d. On the other hand, slower (higher) rate solitons would appear as lines tilted upward (downward) in the plot. In creating the imaging plot, a Hilbert transformation is applied to the interferogram followed by taking the square of its amplitude to produce a pulse envelope intensity profile. The vertical co-rotating time axis can be readily mapped into the soliton angular position within the circular microcavity as shown in Fig. 1d (right vertical axis).

Imaging of soliton formation and multi-soliton trajectories is observable in Fig. 1d. For comparison with the transmitted power, the time-axis scale is identical in Fig. 1c, d. As the pump laser frequency initially scans towards the microcavity resonant frequency its coupled power increases. At ~8 μs the resonator enters the modulation instability regime3,4,17. Initially, a periodic temporal pattern is observable in Fig. 1d corresponding to parametric oscillation51. Soon after, the cavity enters a regime of non-periodic oscillation. At ~31 μs, this regime suddenly transitions into four soliton pulses. The soliton positions evolve with scan time and disappear one-by-one. All solitons have upward curved trajectories, showing that the soliton repetition rate decreases as the scan progresses. This soliton rate shift is caused by the combination of the Raman self-frequency shift effect and anomalous dispersion in the silica resonator42,52 and a similar effect on soliton trajectory is observed in optical fiber resonators43. Supplementary Movie 1 provides the corresponding multi-soliton motion around the microcavity. Finally, the cavity states at four moments in time are plotted within the circular microcavity in Fig. 1e. These correspond to parametric oscillation, non-periodic modulational instability, four soliton, and single soliton states.

Observation of soliton collisions

A variety of non-repetitive multi and single soliton phenomena were measured in both temporal and spectral domains. To enable more rapid imaging the repetition rate of EO comb was adjusted to produce an interferogram at a rate of approximately 50 MHz. The frame period, T, was then reduced accordingly to ~20 ns. Figure 2 presents observations of two solitons interacting. Soliton annihilation is observed in Fig. 2a, wherein two solitons move toward each other, collide, create an intense peak upon collision and then disappear. A new phenomena, a wave splash, is observed immediately following the collision. Though not discussed, this feature appears in recently reported simulations12. In Fig. 2b, two solitons collide but quickly recover and then collide again, after which point one soliton is annihilated. Figure 2c shows a third example in which solitons merge and a single soliton emerges. In a fourth case shown in Fig. 2d, soliton hopping accompanies annihilation of a soliton. Interestingly, all soliton collisions are observed at the beginning of soliton formation (in the soliton breathing regime). After this regime, the soliton relative motion quickly stabilizes preventing collisions. This stabilization process is investigated in a later section. Also, as noted earlier, the observation of these complex motions requires measurement of events in close temporal proximity over long time spans. Finally, numerical simulations of soliton collisions are shown as inset panels in Fig. 2. The collisional features observed in experiments, including the wave splash in Fig. 2a, are reproduced in the simulations.

Breather soliton spectrograms

Figure 3 shows measurement of a breathing soliton39 in both the temporal and frequency domains. The intensity of an individual breather soliton is imaged in Fig. 3a. Spectral breathing was explored in fiber-ring resonators using the DFT method6. In the current work, the spectral breathing is observed by applying a Fourier transform to the interferogram signal47. Figure. 3b shows the resulting spectrogram plotted over the same time interval as Fig. 3a wherein the spectrum is widest when the breather soliton has its maximum peak power. This spectrum also reveals the changing breather period with frequency scan, which has previously been observed by measurement of soliton power26,41. A zoom-in of the soliton temporal breathing is shown in Fig. 3c. The combined high frame rate and sub-ps temporal resolution enable the corresponding amplitude and pulse width of the breather to be extracted and these are plotted in Fig. 3d. As an observation unrelated to the breathing action, the soliton spectral envelope in Fig. 3b is continuously red shifted in frequency by the Raman self-frequency shift18,42 as its average power increases (increasing time in the plot).

Tracking relative soliton motion

Monitoring relative soliton position in real time is important for study of soliton optical memories3,4, their interaction and control5,9 as well as in soliton crystals38. Previously, microcavity soliton relative positions have been measured by autocorrelation53, frequency-resolved optical gating17 and synchronized cross-correlation38. However, with an update rate limited by a mechanical delay line, these methods are only useful for measurement of steady-state phenomena. In this work, relative soliton positions can be measured in real time from the interferogram thereby enabling study of their relative motion dynamics. To plot soliton relative position, one soliton is selected to be the reference (i.e., zero point of the angular position) and the angular position relative to the reference soliton is defined from −π to π. Two representative measurements are shown in Fig. 4a, b wherein the laser frequency is scanned from high to low frequency. Even though the reference soliton round-trip rate is changing as the laser frequency is scanned (see, for example, Fig. 1d), the solitons experience extended stable motion relative to one another. In Fig. 4a, the solitons stabilize relative to each other within a few microseconds after formation and in Fig. 4b, the relative positions are stable from 9 to 22 μs and then destabilize. It is believed that the stabilization of solitons is related to the presence of a dispersive wave caused by an avoided mode crossing54. Note that ultimately, all of the solitons in both panels are annihilated when the laser is tuned beyond the existence detuning range17.

Characterization of soliton decay

Finally, soliton decay is analyzed using the sampling method. The measurement results are shown in Fig. 4c, d. In the experiment, the pump laser frequency is continuously tuned toward lower frequencies. After soliton formation, at some point the cavity-laser frequency detuning exceeds the soliton existence range and the soliton decays17,18. Figure 4c shows the interferogram signal just before and during the decay. Pulse widths (τs) are extracted during the decay process and are plotted in Fig. 4d. Also plotted in Fig. 4d is the product of pulse width and soliton peak amplitude (AE). Curiously, the soliton pulse width and peak amplitude preserve the same soliton product relationship as prior to decay. This is an indication that the decaying soliton pulse in the microcavity is constantly adapting itself to maintain the soliton waveform. A similar behavior is known to occur for conventional solitons in optical fiber55. To the authors knowledge, this is the first time this behavior has been observed in real time. In the Methods section the amplitude decay of the soliton in the interferogram trace is analyzed to extract a decay time and the cavity Q factor.

Importance of high soliton repetition rate

Coherent sampling induces a large bandwidth compression of the ultrafast signal that is equal to the sampling rate divided by the difference in the signal rate and the sampling rate. This compression is well known in the related techniques of dual-comb spectroscopy47 and dual-comb ranging46, and is also present in sampling of optical signals by four-wave mixing in optical fibers56. In order to avoid spectral folding, the compressed signal bandwidth must lie within half of the EO comb sampling rate46,47 (the Nyquist condition for sampling). As shown in the Methods section, this basic condition establishes the following relationship between temporal resolution (τ), frame rate (f) and the sampling rate (approximately the microcavity free-spectral-range, FSR): f<τFSR2/2. This condition also reveals the quadratic importance of high sampling rates (equivalently large FSRs and correspondingly large soliton repetition rates) to create fast frame rates. In the current system, a temporal resolution of <1 ps combined with a 22 GHz sampling rate can enable frame rates as high as 200 MHz.

Numerical simulation of multi-soliton trajectories

Soliton dynamics are governed by the Lugiato-Lefever (LL) equation57 augmented by Raman42,52 and avoided mode crossing58 effects. The LL equation can be simulated numerically using the split-step method55. Simulated intracavity power versus temporal profiles for soliton formation are presented Fig. 5a and, for comparison with the imaging data in Fig. 1d, the corresponding simulated multi-soliton trajectories are plotted in Fig. 5b. In the simulation, the laser frequency is linearly scanned from higher to lower frequency. Moreover, the Raman effect and one avoided mode crossing are included in the simulation. Concerning the vertical axis scale, it is noted that because the periodicity of the soliton interferogram signals varies by <1 % during the scan, the vertical co-rotating time axis can be readily mapped into the soliton angular position axis within the circular microcavity as shown in Fig. 1d, e. The features of soliton formation and evolution observed in Fig. 1d compare well with the numerical simulation.

Discussion

Imaging of nonlinear dynamical phenomena including complex soliton interactions with high temporal/spatial resolution over arbitrary time/length spans has been demonstrated. The temporal resolution in the current experiment is limited to 800 fs, however, resolution at the tens of fs level is possible by spectrally broadening the EO comb59 used for coherent sampling. It is also possible to replace the EO-comb with a microcomb that is closely matched to the FSR of a microcavity to be sampled. Such matching has been used in dual-microcomb spectroscopy demonstrations30,31. In this case, even higher sampling rates would be possible that would enable gigahertz-scale frame rates. Moreover, an implementation of soliton sampling produced within a single microresonator comb has also been recently reported60. The coherent sampling method can serve as a general real-time state visualization tool to monitor the dynamics of microcavity systems. It would provide an ideal way to monitor the formation and evolution of soliton complexes such as Stokes solitons36, soliton number switching37, and soliton crystals38. It can also be used to monitor the state of chip-based optical memories based on microresonator solitons.

Methods

Detailed experimental setup

Figure 6 divides the experimental setup into three sections. In the microresonator section, a tunable, continuous-wave (cw) laser is used to pump the microcavity for production of solitons. An erbium-doped fiber amplifier (EDFA) amplifies its power to 500 mW and an acousto-optic modulator (AOM) is used for rapid control of power to the microcavity. A tunable bandpass filter (BPF) is used to block the spontaneous emission noise from the EDFA. The pump is coupled into the microcavity through a tapered-fiber61,62. The emitted power from the microcavity (along with transmitted pump power) is split by a 90/10 fiber coupler. Ten percent of the power is sent to a fiber-Bragg grating (FBG) filter to separate the pump power and the microcomb power. The drop port output is the pump power transmission, while the through-port output is the comb power. Both the pump transmission and the microcomb power are detected with photodetectors (125 MHz bandwidth). The other 90% of the power is combined with the electro-optic (EO) modulation comb sampling pulse using a second fiber coupler.

In the EO comb setup, a pump laser is amplified by an EDFA to 200 mW and then phase modulated by three tandem lithium niobate modulators. The EO comb and microcavity setup can share the same pump laser when the acousto-optic modulator can provide a frequency offset higher than half of the electrical bandwidth of the interferogram signal (to avoid frequency folding). This is the case in Fig. 1. However, they can also use separate pump lasers, which is demonstrated in Figs. 2 and 3. In this case, the frequency sweep range of the soliton pump laser is always less than the frame rate to ensure that the soliton spectrum can be accurately inferred from Fourier transform of the interferogram. The modulators are driven by amplified electrical signals (frequency close to 22 GHz) that are synchronized by electrical phase shifters. The output power of the electrical amplifiers is 33 dBm. The phase modulated pump is then coupled to an intensity modulator to select only portions of the waveform with a uniform chirp. The intensity modulator is driven by the recycled microwave signal from the external termination port of the first phase modulator. The phase and amplitude of the modulation are controlled by electrical attenuators and phase shifters. A programmable line-by-line waveshaper is used to flatten the EO comb optical spectrum and to nullify the linear chirping so as to form a transform-limited sinc-shaped temporal pulse. The average power from the waveshaper output is around 100 μW. The EO pulses are amplified by an EDFA before combining with the microresonator signal.

In the interferogram measurement, the microcavity signal and the EO pulses are combined in a 90/10 coupler and are then detected by a fast photodetector with 50 GHz bandwidth. An FBG filter is used to block the pump laser of the microcavity to avoid saturation in the photodetector. All photodetected signals are recorded using a 4 GHz bandwidth, 20 GSa/s sampling rate oscilloscope. The center frequencies (compression factors) for the interferograms are around 0.7 GHz (2200) and 2.1 GHz (440) for the 10 MHz and 50 MHz frame rates, respectively.

Time constant in soliton decay

In the soliton decay process, the average intracavity energy decays exponentially and its time constant equals the dissipation rate of the cavity (κ = ω/Q), where ω is the optical frequency and Q is the loaded cavity Q factor. For large cavity-laser frequency detuning17,52, the average intracavity energy is approximately the soliton energy, $$\tau _sA_E^2$$, such that

$$\tau _s(t)A_E^2\left( t \right) = \tau _s\left( 0 \right)A_E^2(0)e^{ - \kappa t}.$$
(1)

When the dissipation rate is relatively small compared to soliton Kerr nonlinear shift, the dissipation is a perturbation and the pulse maintains its soliton waveform55. The corresponding balance of dispersion and Kerr-nonlinearity requires that the product of soliton amplitude and pulse width be constant. This condition was also verified experimentally in Fig. 4d52,55,

$$\tau _s(t)A_E(t) = \tau _s(0)A_E(0).$$
(2)

Inserting eq. (2) into eq. (1) gives,

$$A_E(t) = A_E(0)e^{ - \kappa t},\tau _s(t) = \tau _s(0)e^{\kappa t},A_E^2(t) = A_E^2(0)e^{ - 2\kappa t}.$$
(3)

In particular, the soliton amplitude decays at the cavity dissipation rate, the pulse width exponentially grows, and the soliton peak power decays twice as fast as the cavity dissipation rate. In the experiment, the fitted decay constant of the soliton amplitude is 133 ns, which corresponds to κ/(2π) = 1.2 MHz giving Q = 161 million. This value is in reasonable agreement with the measured loaded-Q factor of 140 million.

Nyquist condition for sampling

In the EO comb sampling process the optical to electrical conversion is accompanied by a large bandwidth compression of the sampled signal. In effect, sampling stretches the time scale so that, for example, the optical temporal resolution (τ) is stretched to τ×FSR/f after conversion to the electrical signal where f is the frame rate given by f ≈ FSR – fcomb. This stretching means that the THz EO comb resolution bandwidth is compressed to an electrical bandwidth of f/(τFSR). To avoid non-sensical signals in the electrical spectrum, the compressed bandwidth should lie within the Nyquist frequency set by the FSR46. This gives the condition f/(τFSR)<FSR/2, or f<τFSR2/2. In practice, when the oscilloscope bandwidth (fosc) is smaller than the Nyquist frequency, the interferogram signal will be limited by the oscilloscope instead of the Nyquist frequency, such that f/(τFSR)<fosc, or f<τfoscFSR. This is, in fact, the case in the present measurement as the oscilloscope bandwidth is 4 GHz while the Nyquist frequency is 11 GHz. In addition, the frequency components of the interferogram signal must be positive to avoid frequency folding near zero frequency. This requires that the carrier frequency of the interferogram signal is larger than half of the electrical bandwidth. In the present measurement, the carrier frequency is the frequency offset between the EO comb pump laser and the microcavity pump laser (defined as ΔΩ). As a result, this condition is expressed as ΔΩ>f/(2τFSR).

Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

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Acknowledgements

The authors thank Stephane Coen and Yun-Feng Xiao for helpful comments during the preparation of this manuscript and gratefully acknowledge the Air Force Office of Scientific Research (AFOSR Award number FA9550-18-1-0353), NASA and the Kavli Nanoscience Institute.

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Contributions

Experiments were designed by all authors. X.Y. and Q.-F.Y. performed the measurements and modeling. K.Y.Y. fabricated devices. X.Y., Q.-F.Y., and K.V. analyzed the data. All authors participated in preparing the manuscript.

Corresponding author

Correspondence to Kerry Vahala.

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Yi, X., Yang, QF., Yang, K.Y. et al. Imaging soliton dynamics in optical microcavities. Nat Commun 9, 3565 (2018). https://doi.org/10.1038/s41467-018-06031-5

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