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Guarantees of Riemannian Optimization for Low Rank Matrix Recovery
Ke Wei, Jian-Feng Cai, Tony F. Chan, Shingyu Leung
TL;DR
Low-rank matrix recovery needs computationally tractable alternatives to nonconvex rank minimization and costly convex methods. This paper analyzes Riemannian gradient and restarted conjugate-gradient algorithms, proving recovery guarantees under restricted-isometry conditions and nearly optimal sampling complexity up to logarithmic factors.
Problem
Low-rank recovery requires scalable alternatives because rank minimization is nonconvex and nuclear-norm methods can be costly for large matrices.
Method
The paper interprets Riemannian optimization algorithms as iterative hard-thresholding methods with projections onto low-dimensional subspaces.
Results
Theoretical guarantees establish recovery for Riemannian gradient descent and restarted conjugate-gradient algorithms, with sampling complexity nearly optimal up to a logarithmic factor.
Takeaways & Limitations
The results support Riemannian optimization as a theoretically guaranteed approach to low-rank matrix recovery under suitable sensing conditions.
Takeaways & Limitations
The main guarantees depend on the measured matrix’s condition number and rank, while a technical subspace choice can fail to capture the full error direction.
Abstract
from arXiv · showhide
We establish theoretical recovery guarantees of a family of Riemannian optimization algorithms for low rank matrix recovery, which is about recovering an $m\times n$ rank $r$ matrix from $p < mn$ number of linear measurements. The algorithms are first interpreted as iterative hard thresholding algorithms with subspace projections. Based on this connection, we show that provided the restricted isometry constant $R_{3r}$ of the sensing operator is less than $C_κ/\sqrt{r}$, the Riemannian gradient descent algorithm and a restarted variant of the Riemannian conjugate gradient algorithm are guaranteed to converge linearly to the underlying rank $r$ matrix if they are initialized by one step hard thresholding. Empirical evaluation shows that the algorithms are able to recover a low rank matrix from nearly the minimum number of measurements necessary.
1 Introduction
The paper studies recovery of low-rank matrices from undersampled linear measurements, motivated by efficient acquisition of high-dimensional data with simple structure. It frames Riemannian optimization as a scalable alternative to nuclear-norm minimization and other direct low-rank methods.
- Problem setting: Low-rank matrix recovery seeks the lowest-rank matrix consistent with p < mn linear measurements of an m×n rank-r matrix.The sensing operator maps matrices to p-dimensional measurement vectors, and entry sensing corresponds to matrix completion.
- Convex relaxation: Nuclear-norm minimization replaces the computationally intractable rank objective with the sum of singular values.Its recovery equivalence with rank minimization can be characterized using the sensing operator’s restricted isometry constant.
- Guarantees: Restricted isometry conditions provide recovery guarantees for rank minimization and nuclear-norm minimization under suitable sensing operators.For example, a sufficiently small rank-5r restricted isometry constant guarantees recovery, with high-probability conditions known for normalized Gaussian and Bernoulli measurements.
- Computational limitations: Interior-point methods for nuclear-norm minimization require linear-system solves for Newton directions, limiting applicability when m and n are large.First-order methods typically use singular value thresholding.
- Approach: The paper studies Riemannian optimization alongside iterative hard thresholding and alternating minimization as algorithms that target low-rank recovery directly.The introduction identifies Riemannian optimization as part of a broader family of nonconvex recovery methods.
2 Algorithms and Main Results
The paper develops Riemannian gradient and restarted conjugate-gradient methods for low-rank recovery by updating within tangent spaces of the rank-r matrix manifold. With one-step hard-thresholding initialization and suitable restricted-isometry conditions, both methods are guaranteed to converge linearly, while requiring nearly minimal measurements up to logarithmic factors.
- Riemannian algorithms: The algorithms avoid full SVD thresholding by restricting search directions to low-dimensional subspaces, reducing the intermediate matrix rank.Full SVD costs O(n^3) when m is proportional to n, motivating subspace-restricted updates.
- Riemannian algorithms: The tangent-space choice turns Algorithms 3 and 4 into Riemannian gradient and conjugate-gradient methods on the rank-r matrix manifold.The tangent space has dimension (m + n − r)r and is defined at the current estimate.
- Recovery guarantees: Under suitable RIC conditions, Riemannian gradient descent and restarted Riemannian conjugate gradient converge linearly from X0 = Hr(A*(y)).Theorems 2.1 and 2.2 establish the guarantees when the corresponding contraction parameter satisfies γ < 1.
- Recovery guarantees: The sufficient condition can be expressed as R3r < Cκ/√r, where Cκ depends on the algorithmic constants.The paper contrasts these conditions with universal-constant RIC conditions and notes that they are more stringent.
- Recovery guarantees: The required number of measurements is nearly optimal up to a logarithmic factor under a random measurement model.The stated sampling complexity follows from high-probability satisfaction of the theorem conditions.
3 Numerical Experiments
Numerical experiments use phase-transition tests to evaluate RGrad, RCG, and restarted RCG under Gaussian and entry sensing. All three algorithms recover rank-r matrices from nearly the minimum number of measurements, with closely related phase-transition behavior.
- Phase transitions: Below the phase-transition curve, the algorithms converge to the measured matrix; above it, they return a mismatching solution.The phase transition is defined over the (δ, ρ) plane.
- Experimental setup: The experiments evaluate RGrad, RCG, and restarted RCG across 18 undersampling ratios δ from 0.1 to 0.95 for Gaussian and entry sensing.Gaussian tests use m = n = 80, while entry-sensing tests use m = n = 800.
- Recovery performance: C · (m + n −r)r measurements, with C slightly larger than 1, suffice for all three tested algorithms to recover rank-r matrices.Tables 1, 2, and Fig. 1 support recovery from nearly the minimum number of measurements.
- Algorithm comparison: RCG and restarted RCG have almost indistinguishable phase-transition curves, while RCG variants slightly exceed RGrad for Gaussian sensing when δ ≥0.6.For entry sensing, RGrad has a slightly higher phase-transition curve.
- Algorithm comparison: The study compares the three Riemannian algorithms with alternating steepest descent on Gaussian and entry-sensing problems.The comparison uses Gaussian instances with m = n = 80, r = 10 and entry-sensing instances with m = n = 8000, r = 100.
4 Proofs of Main Results
The proofs establish convergence by decomposing each iteration’s error into projection, sensing, and manifold-curvature terms, then controlling these terms with restricted-isometry bounds. The conjugate-gradient proof additionally reduces the error dynamics to a three-term recurrence and applies a recurrence-convergence lemma.
- Proof of Theorem 2.1: Theorem 2.1 bounds the next rank-r approximation error by first expanding the projected gradient update and separating it into three terms, I1, I2, and I3.The decomposition follows from X_l+1 being the best rank-r approximation of the intermediate iterate.
- Proof of Theorem 2.1: The I1 bound controls the projected sensing error through the spectral norm of P_Sl − α_l P_Sl A* A P_Sl, using the sensing operator’s restricted-isometry bound and a descent-stepsize bound.The relevant restriction uses rank(P_Sl(Z)) ≤ 2r.
- Initialization: The initialization proof starts from X0 = H_r(A*(y)) and combines subspace projections with restricted-isometry estimates to obtain an initial error bound suitable for induction.The proof defines a 2r-dimensional subspace spanning the column spaces of X0 and X.
- Proof of Theorem 2.2: Theorem 2.2 extends the decomposition to four error contributions, I4 through I7, with the additional term involving the conjugate-gradient coefficient β_l and previous search directions.The bounds reuse the I1–I3 estimates while accounting for the different stepsize and momentum terms.
- Proof of Theorem 2.2: The conjugate-gradient error sequence is rewritten as a three-term recurrence, and Lemma 4.5 yields convergence once its coefficient condition ensures μ < 1.The proof explicitly derives the recurrence for c_l and concludes the first part of Theorem 2.2 by applying Lemma 4.5 and induction.
5 Discussion and Future Direction
The paper identifies condition-number and rank dependencies in its guarantees and outlines future work on removing them. It also proposes extending the algorithms and analysis to other low-rank recovery models, including phase retrieval and blind deconvolution.
- Open limitations: The main recovery guarantees depend on the measured matrix’s condition number and rank.The paper suggests deflation or stagewise techniques as a possible way to eliminate the condition-number dependence.
- Open limitations: Removing the dependence on √r may require analyzing convergence in the matrix operator norm rather than the Frobenius norm.Whether iterates contract under the matrix operator norm remains an open question.
- Future analysis: Future analysis includes other choices of β_l and different Riemannian optimization algorithms beyond the restarted conjugate-gradient variant.The discussed restarted variant uses the selection of β_l developed in.
- Future applications: The algorithms also apply to low-rank recovery models such as phase retrieval and blind deconvolution, where lifting produces a rank-one matrix.The paper identifies this extension as an independent future research direction.
A Proof of Lemma 4.2
The section proves the left inequalities in (23) and (24), while noting that the right inequalities follow similarly. The argument uses direct calculations, a standard textbook result, and the identity (I − U_lU_l^*)X_l = 0.
- The proof establishes the left inequality of (23) through direct calculations.
- The derivation invokes a standard textbook result and the fact that (I − U_lU_l^*)X_l = 0.
- The proof of the left inequality of (24) begins by establishing an intermediate statement, including equality (49).Equality (49) is obtained by noting an additional relation.
B Proofs of Lemmas 4.3 and 4.4
The section proves Lemmas 4.3 and 4.4 using restricted-isometry bounds, normalization, the parallelogram identity, and rank-based inequalities.
- Proof of Lemma 4.3: Lemma 4.3 is proved by adapting the compressed-sensing argument of [14, Lem. 2.1], normalizing both matrices to unit Frobenius norm, and applying RIC bounds.The proof then invokes the parallelogram identity to complete the argument.
- Proof of Lemma 4.4: Lemma 4.4 uses Lemma 4.3 together with rank((I − P_Sl)(X)) ≤ r and rank(P_Sl(Z)) ≤ 2r.These rank bounds justify the second-to-last inequality in the proof.