Exact Matrix Completion via Convex Optimization

Exact Matrix Completion via Convex Optimization Emmanuel J. Candès Benjamin Recht

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Exact Matrix Completion via Convex Optimization 2009 Foundations of Computational Mathematics / arXiv 10.1007/s10208-009-9045-5 0805.4471 candes2009-matrix-completion candes2009-matrix-completion.txt, the project's staged copy 2689bd96b52b634c7f94cb6dc3c834c506210102 arxiv-open mathematics

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0 spans located by searching the stored text for the quote's exact words, allowing any whitespace between them method Quotation withheld because this record is not cleared for redistribution. low-rank matrix completion Quotation withheld because this record is not cleared for redistribution. mathematical-proof mathematical-proof Quotation withheld because this record is not cleared for redistribution. mechanisms the quantity actually wanted - the rank - is replaced by the sum of the singular values, because minimising rank is NP-hard and every known exact algorithm takes time doubly exponential in the dimension, while the substitute is convex Quotation withheld because this record is not cleared for redistribution. mechanisms the substitution is licensed by an analogy the authors state rather than assume: the nuclear norm stands to the rank function as the convex l1 norm stands to the counting l0 norm in sparse signal recovery - one counts nonvanishing singular values, the other sums their amplitude about this quote: 2026-08-16: TWO FAILED GLYPHS INSIDE THIS QUOTE. The staged text reads "what the convex (cid:4)1 norm is to the counting (cid:4)0 norm" - the extraction could not map the script-l character and left the raw PDF font code "(cid:4)" in its place, twice. Read as l-1 and l-0, the standard sparse-recovery notation. The quote is an exact substring of the staged file. It is kept rather than dropped because this sentence IS the licence for the paper's central substitution - the nuclear norm stands to rank as the convex norm stands to the counting norm - and a mechanisms layer that silently omitted it would be tidier and less true. Quotation withheld because this record is not cleared for redistribution. mechanisms the informal requirement that the singular vectors be spread out is converted into a MEASURED quantity - the coherence of a subspace against the standard basis - whose smallest value is one, attained when the spanning vectors' entries all share a magnitude, and whose largest is attained by any subspace containing a standard basis element Quotation withheld because this record is not cleared for redistribution. mechanisms necessity is established by exhibiting a matrix rather than by argument: this one cannot be recovered from a sample of its entries unless nearly all are seen, because for most sampling sets one would see only zeros and would have no way to guess the matrix is not zero about this quote: 2026-08-16: THE SUBJECT OF THIS QUOTE IS ANAPHORIC AND ITS REFERENT IS NOT QUOTABLE. "this matrix" is the rank-one matrix with a single nonzero entry in the top-right corner, defined in the preceding sentence as M = e1 e_n^*. The span was NOT widened to reach that definition because the intervening text is the flattened matrix display, which the extraction renders as stray zeros and box-drawing characters - 'M = e1e* n = [ 0 0 ... 0 0 . . . 0 . . . ] , 0 1 0 0 ...' - so a wider span would quote rubble to fix a pronoun. Recorded here instead. Quotation withheld because this record is not cleared for redistribution. mechanisms a second necessary condition is derived the same way, from an adversarial sampling set rather than a random one: with no sample in some row, one component is unobserved and no method whatsoever can recover it, so at least one observation per row and per column is required Quotation withheld because this record is not cleared for redistribution. mechanisms the impossibility is side-stepped by changing the question from ALL sampling sets to MOST - adversarial sampling defeats even rank one, so what is asked instead is whether recovery succeeds for almost all sampling sets of a given size Quotation withheld because this record is not cleared for redistribution. mechanisms the class of matrices the theorem covers is fixed by an explicitly stated assumption rather than left to the reader: both coherences must be bounded above by a single positive constant, which is what turns 'the singular vectors are spread out' into a checkable premise Quotation withheld because this record is not cleared for redistribution. mechanisms the convex program is put into a form existing software already solves: for a positive semidefinite unknown the nuclear norm IS the trace of the matrix, so the heuristic collapses into minimising the trace over the constraint set Quotation withheld because this record is not cleared for redistribution. The guarantee covers exactly low-rank matrices; extension to approximately low-rank matrices is left open. what this limits: the exact-recovery guarantee: IT WOULD BE OF INTEREST TO EXTEND OUR RESULTS TO THE CASE WHERE THE UNKNOWN MATRIX IS APPROXIMATELY LOW-RANK - only exactly low-rank is covered Quotation withheld because this record is not cleared for redistribution. The sampling bounds proved may be far more demanding than what is needed in practice. what this limits: the sampling bound as a practical requirement: THE EXPERIMENTS REVEAL VERY SIMILAR PLOTS FOR DIFFERENT n, SUGGESTING THAT OUR ASYMPTOTIC CONDITIONS FOR RECOVERY MAY BE RATHER CONSERVATIVE Quotation withheld because this record is not cleared for redistribution. The semidefinite program the theory relies on does not scale with off-the-shelf solvers. what this limits: any large-scale application of the method as specified: STANDARD INTERIOR POINT METHODS WILL BE IMPRACTICAL FOR PROBLEMS WHERE THE UNKNOWN MATRIX HAS HUNDREDS OF ROWS OR COLUMNS Quotation withheld because this record is not cleared for redistribution. The general problem of recovery from arbitrary linear functionals is not treated. what this limits: the general recovery problem: WE LEAVE THIS GENERAL LOW-RANK RECOVERY PROBLEM FOR FUTURE WORK Quotation withheld because this record is not cleared for redistribution. The paper states that the general problem is impossible and narrows the question before answering it. what this limits: the exact-recovery result reported here: the authors state it is impossible to recover all low-rank matrices from sampled entries, and pose the answerable question as whether most can be recovered Quotation withheld because this record is not cleared for redistribution. The tractable method is a relaxation whose equivalence to the real problem holds for most instances, not all. what this limits: the use of nuclear norm minimisation as a stand-in for rank minimisation: the paper states the relaxation is formally equivalent to the combinatorially hard rank minimization problem for most problems Quotation withheld because this record is not cleared for redistribution. The recovery guarantee holds only for matrices whose row and column spaces satisfy an incoherence condition. what this limits: any application of this recovery result: it applies to matrices whose row and column spaces are incoherent with the standard basis, a property of the matrix that must hold for the guarantee to bind Quotation withheld because this record is not cleared for redistribution. The exact-solution alternative is not merely slow but doubly exponential in the dimension. what this limits: the claim that this method makes the problem tractable: the exact formulation is NP-hard and all known exact algorithms require time doubly exponential in the matrix dimension Quotation withheld because this record is not cleared for redistribution. the compressed-sensing result that an object can be recovered exactly from far fewer measurements than its dimension, carried over from signal acquisition to the recovery of a low-rank matrix from a sample of its entries Quotation withheld because this record is not cleared for redistribution. the whole recovery procedure repeated 50 times for every (n, m, r) triple eliminates: a single lucky or unlucky draw of the sampled entries - 50 repetitions per parameter setting turn one outcome into a rate · outcome (stated by the tagger, no quotation): held Quotation withheld because this record is not cleared for redistribution. nuclear norm nuclear norm Quotation withheld because this record is not cleared for redistribution. incoherent incoherent Quotation withheld because this record is not cleared for redistribution. standard basis standard basis Quotation withheld because this record is not cleared for redistribution. rank minimization rank minimization Quotation withheld because this record is not cleared for redistribution. sampling operator sampling operator Quotation withheld because this record is not cleared for redistribution. 'exact' recovery - EXACT UNDER CONDITIONS, NOT IN GENERAL: recovery of the true matrix is proved for matrices incoherent with the standard basis and from a random sample, after the paper has stated that recovering ALL low-rank matrices is impossible, so the word describes the outcome when the hypotheses hold rather than the reach of the method Quotation withheld because this record is not cleared for redistribution. the nuclear norm - A HEURISTIC STANDING IN FOR RANK: the problem actually wanted is combinatorially hard, the nuclear norm is introduced as a heuristic optimisation, and the theorem is that the two coincide for most problems - so the method solves a different problem that usually has the same answer Quotation withheld because this record is not cleared for redistribution. incoherence - THE HYPOTHESIS THE HEADLINE DEPENDS ON: the recoverable matrices are those whose row and column spaces are incoherent with the standard basis, a property nothing in an application guarantees and which the result cannot be quoted without Quotation withheld because this record is not cleared for redistribution. builds-on → convex optimization (nuclear-norm minimization) Quotation withheld because this record is not cleared for redistribution. extends → the theory of compressed sensing Quotation withheld because this record is not cleared for redistribution. builds-on → the nuclear norm heuristic introduced by Fazel [19] Quotation withheld because this record is not cleared for redistribution. extends → the notion of incoherence of a unitary transformation introduced in [9] Quotation withheld because this record is not cleared for redistribution. nuclear norm —same-as→ rank minimization nuclear norm rank minimization holds when: for most problems, not for all · strength of the relation: 0.9 Quotation withheld because this record is not cleared for redistribution. m ≥ C·n^1.2·r·log n — sample complexity: entries sufficient to recover most rank-r n×n matrices Quotation withheld because this record is not cleared for redistribution. 1.25 — replacing 1.2 with 1.25 makes the result hold for all ranks Quotation withheld because this record is not cleared for redistribution. matrix completion matrix completion Quotation withheld because this record is not cleared for redistribution. nuclear norm minimization nuclear norm minimization Quotation withheld because this record is not cleared for redistribution. chi-square chi-square Quotation withheld because this record is not cleared for redistribution. degrees-of-freedom degrees-of-freedom Quotation withheld because this record is not cleared for redistribution. sample-size sample-size about this quote: QUOTE RECOVERED FROM THE PAPER'S OWN CHARACTERS. The recorded quote was LETTERS-ONLY IDENTICAL to this stretch of the staged text but not an exact substring of it - spaces, hyphens, ligature glyphs or mathematical operators differed, so it could never anchor. This is the file's own text between the same two letters: not produced by a rule about the damage, copied out of the file. Round-trip checked (the recovered span is letters-only equal to the original) and length-checked. The text as originally recorded is kept in evidence as extracted. Quotation withheld because this record is not cleared for redistribution.

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