Recovering low-rank matrices from few coefficients in any basis

David Gross

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Recovering low-rank matrices from few coefficients in any basis

David Gross

arXiv:0910.1879v5Oct 12, 2009cs.ITmath.NAquant-ph

TL;DR

The paper asks how to efficiently recover a low-rank matrix from a small number of expansion coefficients. It develops simpler, more general techniques based on trace-norm recovery and incoherence, obtaining guarantees for arbitrary operator bases. For a rank-r matrix with coherence ν, O(nrν(1+β) ln^2 n) sampled coefficients suffice for unique recovery with probability at least 1−n^−β.

Problem
The central problem is whether a low-rank matrix can be recovered computationally efficiently from a limited number of expansion coefficients.
Method
The paper combines trace-norm convex optimization with incoherence-based analysis for randomly sampled expansion coefficients and arbitrary operator bases.
Results
O(nrν(1+β) ln^2 n) sampled coefficients suffice for unique recovery with probability at least 1−n^−β.
Takeaways & Limitations
The approach extends low-rank recovery analysis beyond matrix elements to arbitrary operator bases while improving the ν-dependency and simplifying the proof.

Abstract

from arXiv · show

We present novel techniques for analyzing the problem of low-rank matrix recovery. The methods are both considerably simpler and more general than previous approaches. It is shown that an unknown (n x n) matrix of rank r can be efficiently reconstructed from only O(n r nu log^2 n) randomly sampled expansion coefficients with respect to any given matrix basis. The number nu quantifies the "degree of incoherence" between the unknown matrix and the basis. Existing work concentrated mostly on the problem of "matrix completion" where one aims to recover a low-rank matrix from randomly selected matrix elements. Our result covers this situation as a special case. The proof consists of a series of relatively elementary steps, which stands in contrast to the highly involved methods previously employed to obtain comparable results. In cases where bounds had been known before, our estimates are slightly tighter. We discuss operator bases which are incoherent to all low-rank matrices simultaneously. For these bases, we show that O(n r nu log n) randomly sampled expansion coefficients suffice to recover any low-rank matrix with high probability. The latter bound is tight up to multiplicative constants.

I. INTRODUCTION

The paper develops efficient low-rank matrix recovery from randomly sampled expansion coefficients, extending prior matrix-completion results to arbitrary operator bases. Its analysis uses incoherence and simpler, more general proof techniques to establish recovery guarantees.

Motivation: Roughly rn parameters specify an n×n matrix of rank r, but computationally tractable recovery from that many coefficients is not straightforward.
Motivation: Low-rank matrix recovery is presented as a non-commutative analogue of compressed sensing, replacing vector sparsity with matrix low rank.
Contribution: The paper builds on earlier methods to provide a more accessible treatment, arbitrary operator-basis results, fuller proofs, and tighter estimates.
Method: The recovery algorithm minimizes the trace norm subject to matching the revealed expansion coefficients, and Theorem 3 relates uniqueness to the number of coefficients sampled.The optimization is efficiently implementable over the space of matrices.
Incoherence: Incoherence ensures that typical expansion coefficients contain enough non-trivial information about the unknown matrix; some bases are incoherent with every low-rank matrix.
Contribution: The main theorem applies to arbitrary operator bases, improves the ν-dependency, and is easier to prove than earlier results.
Guarantees: O(nrν(1+β) ln^2 n) sampled coefficients suffice for unique recovery with probability at least 1−n^−β for a rank-r matrix with coherence ν.

B. Examples

The paper applies its recovery framework to matrix completion and operator bases, including Pauli bases used in quantum-state tomography. It explains trace-norm recovery geometrically through affine constraints and dual certificates.

Matrix completion is treated as a special case of recovery from sampled expansion coefficients in the standard operator basis.
ν = 1 is the best possible operator-norm parameter, achieved when √n w_a is unitary for every basis element.
The Pauli operator basis is a tensor-product basis with applications in experimental quantum-state tomography.
A dual certificate proves recovery by simultaneously supporting the trace-norm ball at ρ and being normal to the affine constraint space.
Trace-norm minimization provides an efficiently solvable proxy for the NP-hard problem of minimizing rank over the affine space of matrices matching observed coefficients.

D. Novel approaches

The paper develops a simpler analysis based on independent sampling, operator-valued concentration, and approximate dual certificates. These tools extend recovery guarantees beyond matrix elements to arbitrary operator bases.

The paper replaces earlier basis-specific analyses with an i.i.d. sampling model that is easier to analyze because sampled variables are independent and identically distributed.
Operator-valued large-deviation bounds are used at multiple stages of the proof to control random matrix observables.
The dual-certificate construction requires PTY to be close to sgn ρ rather than exactly equal, simplifying the certificate conditions.
The semidefinite program minimizes the trace norm subject to matching the observed coefficients, with sampling replacement affecting collisions and effective information.
A recursive random process, together with operator Bernstein concentration, constructs certificates more simply than previous approaches.

C. First case: large ∆T

When the tangent component of a deviation is large, concentration bounds show that the deviation is unlikely to satisfy the sampling constraints. The analysis uses operator Bernstein inequalities to control the relevant random operators.

The first proof case establishes infeasibility when ∥∆_T∥_2 is much larger than ∥∆_T⊥∥_2.
The sampling operator’s norm is controlled through the maximum collision count among independently sampled basis indices.
E[P_T R P_T] = P_T, so the analysis bounds deviations of P_T R P_T from its expectation in operator norm.
A simpler n^2 estimate is retained because the coefficient enters the final sample bound only logarithmically.
The operator-Bernstein inequality bounds sums of independent, zero-mean, Hermitian matrix-valued random variables using variance and norm parameters.
For n = 1, the operator inequality reduces to the standard Bernstein inequality.

E. Second case: small ∆T

When the tangent component is small, recovery follows from constructing a dual certificate in the sampling operator’s range. The certificate forces every feasible nonzero deviation to increase the trace norm.

A certificate Y in range R with controlled tangent and orthogonal components ensures that feasible deviations increase the trace norm.
The certificate argument proves that the optimization solution is uniquely equal to ρ.
The certificate construction is simplified using operator Bernstein concentration and a recursive random process that converges quickly to Y.
The paper introduces an initial ansatz for constructing the dual certificate before developing the improved recursive construction.

1) Intuition:

The golfing scheme builds a dual certificate sequentially by repeatedly reducing the residual toward the origin. Independent batches support exponential convergence while balancing batch size against the number of iterations.

1) Intuition:: Batch size must be large enough for operator concentration bounds but small enough to preserve exponential improvement across l = m/k iterations.This creates a trade-off between per-batch control and the number of refinement steps.
1) Intuition:: The golfing scheme adaptively approximates each residual and reduces the next goal’s distance to the origin by half.The residual sequence therefore converges exponentially fast.
1) Intuition:: Each iteration samples a batch of basis elements and uses its sampling operator to update the certificate approximation.The construction samples l batches, with the ith batch containing mi = κirn matrices.
1) Intuition:: The construction controls approximation and failure events across iterations, then combines their probabilities using a union bound.The resulting conditions ensure the golfing updates succeed simultaneously with controlled failure probability.
1) Intuition:: For the analyzed Fourier-type setting, the sampled-coefficient requirement scales as O(rnν(1 + β) ln^2 n).The golfing construction is sequential, unlike earlier holistic certificate construction by direct inversion.

3) Discussion:

The discussion identifies ways to generalize and sharpen the certificate analysis while noting limitations from non-uniform bounds and dependencies in more ambitious sequential schemes. The construction extends beyond small operator-norm bases through incoherence assumptions.

3) Discussion:: Choosing each coefficient optimally could minimize the residual distance, but it introduces strong dependencies that defeat the paper’s elementary i.i.d. analysis.Whether martingale bounds suffice to analyze that optimal scheme remains unclear.
3) Discussion:: The golfing scheme can reuse one fixed set of O(nr ln n) basis elements, but the proof’s non-uniform bounds motivate using different sets across runs.A sharper substitute for the crude union bound could potentially remove this less-optimal feature.
3) Discussion:: The analysis can replace uniform operator-norm estimates with non-uniform bounds on each residual, yielding bounds of order O(rnν ln n).The golfing scheme only requires control of (PT RiPT − 1)Xi for the current residual Xi.
3) Discussion:: The sequential construction is formally analogous to a Neumann-series certificate, but independent sampling operators make its product structurally different.The added independence is described as making the product more benign than repeated powers of one operator.
3) Discussion:: The certificate construction extends from bases with small operator norm to arbitrary operator bases satisfying incoherence properties.The general argument controls overlaps between residual matrices and basis elements through µ(F).
3) Discussion:: The general-case proof imposes alternative overlap bounds on matrices in the tangent space and propagates them through the golfing iterations.These bounds support concentration estimates for the residual-dependent updates.

III. REFINED METHODS AND GENERALIZATIONS

The paper develops dimension-free and refined probabilistic tools for low-rank recovery, including a sharper analysis of the golfing scheme for Fourier-type bases. These refinements yield improved coefficient bounds, while the general case retains a union-bound limitation.

A. Martingale methods for matrix-valued random variables: The section introduces a dimension-free vector bound and an operator martingale variance bound to strengthen the recovery analysis.The operator estimate generalizes the operator Chernoff bound and supports tighter recovery bounds.
B. Tighter bounds for Fourier-type bases: The refined golfing scheme combines dimension-free large-deviation estimates with more carefully chosen random-process parameters.The changes target tighter bounds for Fourier-type operator bases.
B. Tighter bounds for Fourier-type bases: The resulting theorem guarantees unique recovery with failure probability below e−β under a Fourier-type operator-norm condition.The theorem states the guarantee for a rank-r matrix and an operator basis satisfying the displayed incoherence bound.
B. Tighter bounds for Fourier-type bases: The analysis removes an unnecessary (ln ln n)-term by retaining only batches whose assumptions hold and compensating with additional sampled batches.This construction introduces a larger batch pool and selects an admissible subset through an indexing function.
B. Tighter bounds for Fourier-type bases: The general-case analysis does not state O(rnν ln n) bounds because a union bound in Lemma 10 remains the sole obstruction.The author considers overcoming this difficulty plausible but unresolved.

C. A lower bound

The lower-bound section shows that the tighter Fourier-type recovery bound cannot generally be improved beyond multiplicative constants. Pauli-basis constructions provide both deterministic and probabilistic non-uniqueness examples.

C. A lower bound: The lower-bound results imply that the bound of Theorem 4 is tight up to multiplicative constants.This establishes order-level optimality for the Fourier-type recovery guarantee.
C. A lower bound: For the Pauli basis, fewer than (n −2) log2 n sampled coefficients can leave two orthogonal rank-one projections indistinguishable.The matching coefficients occur for every sampled basis element in the set.
C. A lower bound: A rank-one projector also admits a compatible orthogonal projector with nontrivial probability under repeated random sampling.The construction analyzes samples drawn with replacement from the Pauli basis.
C. A lower bound: The Pauli-basis proof uses stabilizer groups and characters to construct compatible orthogonal projectors.The argument finds a stabilizer group with a small intersection with the sampled set, then uses distinct characters agreeing on that intersection.

D. Non-Hermitian setting

The paper extends its Hermitian recovery results to arbitrary complex matrices by embedding them into a larger Hermitian problem. The resulting bounds remain valid after replacing n and r by 2n and 2r.

D. Non-Hermitian setting: A complex n × n matrix is mapped to a Hermitian 2n × 2n matrix, enabling the existing Hermitian recovery analysis to be applied.The construction builds a corresponding Hermitian basis and recovery problem in the extended space.
D. Non-Hermitian setting: The extension computes the embedded matrix’s incoherence properties and relates its recovery program back to the original problem.The proof proceeds through basis construction, coherence analysis, extended recovery, and comparison of failure behavior.
D. Non-Hermitian setting: The embedded matrix has a structured singular-value and eigenspace representation supporting the Hermitian reduction.The construction uses normalized eigenvectors of the Hermitian embedding associated with the nonzero singular directions.
D. Non-Hermitian setting: Theorem 3 and Theorem 4 extend to non-Hermitian matrices when coherence is defined accordingly, with n and r replaced by 2n and 2r.The extended optimization has the embedded matrix as its unique solution, and trace-norm proportionality transfers uniqueness back.

IV. CONCLUSION AND OUTLOOK

The outlook identifies noise robustness and tight frames as extensions of the recovery framework, while noting that sampling with replacement can worsen the sampling-operator norm. Continuous frames remain outside the direct scope of one key lemma.

IV. CONCLUSION AND OUTLOOK: The procedures are described as resilient against noise, but the general noise analysis is deferred to techniques from matrix completion.The relevant bounds are sensitive to the operator norm of the sampling operator.
IV. CONCLUSION AND OUTLOOK: Sampling with replacement can make the sampling-operator norm likely O(ln n), whereas sampling without replacement gives norm one.This difference affects the sensitivity of noise bounds.
IV. CONCLUSION AND OUTLOOK: Tight frames can replace orthonormal bases in the Fourier-type case when their elements have a uniform operator-norm bound.Under that condition, the paper’s statements translate readily from orthonormal bases to tight frames.
IV. CONCLUSION AND OUTLOOK: Without a uniform frame bound, Lemma 10’s union bound depends on frame size and cannot be directly applied to continuous frames.The author believes this difficulty may be overcome but leaves further details for future work.

VI. APPENDIX A: PROOF OF THEOREM 12

The appendix proves Theorem 12 using a Doob-martingale argument, then develops stabilizer groups from Pauli operators and their associated rank-one projectors.

VI. APPENDIX A: PROOF OF THEOREM 12: Theorem 12 is proved by applying Theorem 11 to a Doob martingale sequence and bounding its differences with conditional expectations.The proof sets X0 = 1 and Di = Zi − Zi−1, then uses the triangle inequality and an expectation bound.
Pauli operators: Pauli operators multiply up to phases, and commuting operators simplify this product relation while forming the Pauli group.The operators are Hermitian and unitary, and the phase is real when commuting products remain Hermitian.
Stabilizer states: Stabilizer groups are Abelian subgroups of the Pauli group that exclude −1 and have order 2^k.These conditions define the groups used to construct the projection operators.
Stabilizer states: For a stabilizer group and a complex character, the constructed operator has trace one and is idempotent, making it a rank-one projector.The proposition establishes the projector properties through character orthogonality and Hermiticity of the Pauli operators.
Stabilizer states: A finite-field construction generates stabilizer groups Gx whose elements have the form ±w(p, xp), with exactly one sign included for each p.The construction uses a basis of F_2^k and the identification with the finite field of order 2^k.
Stabilizer states: Every stabilizer group is isomorphic to the additive group of F_2^k.This final proposition characterizes the group structure underlying the construction.

2. Given any l < k elements {g1, . . . , gl}

Given fewer than k elements of a stabilizer group, distinct characters can agree on those elements because the corresponding subspace has dimension below k.

2. Given any l < k elements {g1, . . . , gl}: Two distinct characters can agree on g1, . . . , gl when l < k.The cited result concerns partial agreement on the specified group elements.
2. Given any l < k elements {g1, . . . , gl}: The elements g1, . . . , gl span a subspace of F_2^k with dimension at most l.The dimension bound follows directly from having l spanning elements.
2. Given any l < k elements {g1, . . . , gl}: There are 2^(k−l) ways to extend a functional from the l-dimensional subspace to all of F_2^k.Characters of F_2^k correspond one-to-one with linear functionals to F_2.