Recovering a matrix instead of a vector

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

It is known that given corrupt measurements $y = Af+e$ one can recover an input vector $f \in \textbf{R}^n$ exactly by solving a convex optimization problem. What if $f$ is instead a square matrix? Can we recover a matrix from corrupt measurements instead of just a vector?

-

7
Isn't a matrix just a special case of a vector? That is, can't you "flatten" an $n\times n$ matrix into a vector in $\mathbb{R}^{n^2}$ and then solve the problem by the usual means? – Noah Stein Nov 23 2011 at 3:00

1
I'm not sure I agree with the assertion in the first sentence. It seems to depend strongly on the nature of corruption, and on the precision of measurements. – S. Carnahan♦ Nov 23 2011 at 5:31

1 Answer

First, I emphasize S. Carnahan's comment: Exact recovery from noisy measurements is not that simple. "Exact recovery" usually means "recovery of the exact support of $f$". Moreover, sparsity assumptions for $f$ and special assumptions for $A$ and the size of $e$ are needed.

To address your question: This again depends on a lot of things. Of course you can view this as $n$ multiple instances of the original problem and basically use the previous theory. Other structural assumption lead to other results (e.g. having a "joint sparsity pattern in the columns of $f$"). Buzzwords here are "joint sparsity" or "multiple measurement vectors".

-
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 11, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9185693860054016, "perplexity_flag": "middle"}