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Consistency of trace norm minimization

Francis Bach

arXiv:0710.2848v1cs.LG

TL;DR

Trace-norm regularization is used to estimate low-rank rectangular matrices, but its rank-selection consistency requires characterization. The paper extends Lasso and group-Lasso theory to derive necessary and sufficient conditions for square-loss rank consistency, then develops an adaptive version. The adaptive procedure achieves n^-1/2-consistency and rank consistency without consistency conditions.

  • Problem

    The paper addresses whether trace-norm minimization with square loss can consistently estimate a low-rank matrix and its rank.

  • Method

    The paper extends Lasso and group-Lasso consistency analysis to trace-norm regularization and designs an adaptive trace-norm procedure.

  • Results

    The adaptive version achieves n^-1/2-consistency and rank consistency without consistency conditions.

  • Takeaways & Limitations

    Adaptive trace-norm regularization can provide rank consistency even when the necessary condition for the non-adaptive procedure is not fulfilled.

  • Takeaways & Limitations

    The paper leaves the limiting case ∥Λ∥2 = 1, requiring more complex necessary and sufficient conditions, for future research.

Abstract

from arXiv · show

Regularization by the sum of singular values, also referred to as the trace norm, is a popular technique for estimating low rank rectangular matrices. In this paper, we extend some of the consistency results of the Lasso to provide necessary and sufficient conditions for rank consistency of trace norm minimization with the square loss. We also provide an adaptive version that is rank consistent even when the necessary condition for the non adaptive version is not fulfilled.

1. Introduction

The paper extends sparse-regression consistency analysis to trace-norm regularization for estimating the rank of low-rank rectangular matrices. It derives conditions for rank consistency and introduces an adaptive procedure that avoids the non-adaptive condition.

  • The trace norm extends ℓ1 regularization from vectors to rectangular matrices and typically produces low-rank solutions.It is the sum of singular values and the convex envelope of rank on the spectral-norm unit ball.
  • The paper asks whether trace-norm minimization with square loss consistently estimates a matrix and its rank when the data come from a low-rank matrix.
  • Necessary and sufficient rank-consistency conditions extend corresponding results for the Lasso and group Lasso under full i.i.d. and collaborative-filtering sampling assumptions.
  • The adaptive trace-norm procedure achieves n^-1/2-consistency and rank consistency without consistency conditions.The non-adaptive procedure may fail to estimate rank correctly when its necessary condition is not satisfied.

2. Notations

The notation section defines the vector and matrix norms used throughout the paper, along with matrix-vectorization operations and asymptotic probability notation. It also establishes conventions for vectors, matrices, and population quantities.

  • Vectors use the Euclidean norm, while matrices use spectral, trace, and Frobenius norms based on their singular values.The spectral norm is the largest singular value, the trace norm is their sum, and the Frobenius norm is their ℓ2-norm.
  • For a matrix M, vec(M) stacks columns into a vector, and A ⊗ B denotes the Kronecker product.The section records identities connecting vectorization, matrix multiplication, and Kronecker products.
  • The notations Op(an) and op(an) describe stochastic order and convergence to zero in probability, respectively.
  • Lowercase letters denote vectors, uppercase letters denote matrices, and bold fonts denote population quantities.

3. Trace norm minimization

The paper formulates square-loss prediction with trace-norm regularization, relates the framework to Lasso, group Lasso, and collaborative filtering, and develops assumptions and optimality conditions for analysis.

  • Trace norm minimization: The prediction problem estimates a matrix W from observations (Mi, zi) using a linear model and square-loss optimization.The target matrix is assumed nonzero and rank-deficient, so both the matrix and its rank are estimated.
  • 3.1 Special cases: Trace-norm regularization generalizes Lasso and group Lasso by allowing singular vectors to vary and act as implicitly learned features.For diagonal matrices it reduces to ℓ1 regularization; for block-diagonal matrices it becomes a sum of group norms.
  • 3.1 Special cases: Collaborative filtering represents bilinear prediction as a linear form in rank-one matrices, but growing matrix dimensions place low-rank completion outside the paper’s scope.
  • 3.2 Assumptions: The analysis uses full i.i.d. and non-i.i.d. sampling assumptions, including moment, covariance, and asymptotic normality conditions.The non-i.i.d. setting models uniformly sampled pairs of independently generated x and y attributes.
  • 3.3 Optimality conditions: The trace-norm optimality condition characterizes the fitted matrix through its singular vectors and requires aligned singular-value conditions for the residual term.In the diagonal case, these conditions reduce to the usual Lasso optimality conditions.

4. Consistency results

The paper characterizes how the regularization rate determines regular and rank consistency for trace norm estimation. Rank consistency requires specific conditions in the slow-decay regime, while adaptive regularization can recover consistency without the nonadaptive condition.

  • If λn does not tend to zero, the trace norm estimate is not consistent.
  • If λn tends to zero faster than n^-1/2, the estimate is consistent with error Op(n^-1/2) but overestimates rank with probability tending to one.
  • If λn tends to zero at rate n^-1/2, the estimator has error Op(n^-1/2), while correct-rank probability converges to a limit strictly between zero and one.
  • If λn tends to zero more slowly than n^-1/2, the estimate has error Op(λn), and rank consistency depends on specific consistency conditions.
  • When n^1/2λn tends to infinity and λn tends to zero, the stated sufficient condition yields both consistency and rank consistency, while the stated necessary condition is required for any sequence achieving both.
  • The necessary condition is not generally sufficient for the group Lasso, and the limiting case ∥Λ∥2 = 1 requires additional conditions left for future research.
  • The paper proves whether correct-rank probability tends to zero or one, while convergence rates require additional assumptions under investigation.
  • The analysis uses full i.i.d. and non-i.i.d. sampling assumptions, with extensions allowing invertibility of H⊤ΣmmH and relating ∥Λ∥2 ≤ 1 to the group-Lasso condition.

5. Adaptive version

The adaptive trace-norm procedure replaces the unweighted penalty using a least-squares estimate and achieves consistency and rank consistency without the non-adaptive consistency conditions. Under its tuning assumptions, the estimator is also asymptotically normal.

  • Adaptive weighting: The least-squares estimator ˆWLS provides the basis for constructing the adaptive penalty weights.Its singular-value decomposition is completed by n^-1/2 values to reach dimensions p and q.
  • Adaptive weighting: The adaptive procedure replaces ∥W∥* with ∥AWB∥*, using two positive definite symmetric matrices, paralleling the adaptive Lasso.In the Lasso special case, this replacement exactly corresponds to Zou’s adaptive Lasso.
  • Consistency: Under γ ∈ (0, 1], n^1/2λn → 0, and λnn^(1/2+γ/2) → ∞, any global minimizer ˆWA is consistent and rank consistent.These conditions yield rank consistency without conditions such as the non-adaptive requirements in Eq. (7) or Eq. (8).
  • Asymptotic behavior: The adaptive estimator satisfies n^1/2 vec(ˆWA−W) convergence in distribution to a zero-mean normal distribution with a stated covariance matrix.The asymptotic result accompanies the consistency and rank-consistency guarantees.
  • Scope of the theorem: The restriction γ ≤ 1 arises because ˆWLS estimates the singular subspaces only at rate Op(n^-1/2).Synthetic examples illustrate the theorem and exhibit singular behavior at the limiting case γ = 1.

6. Algorithms and simulations

The paper develops a smoothed trace-norm optimization algorithm and uses simulations to illustrate rank-consistency theory for regular and adaptive penalties.

  • 6.1 Smoothing: The algorithm smooths the nondifferentiable trace norm by adding a strictly convex spectral barrier to the dual problem.The barrier depends on singular values and produces a differentiable primal formulation.
  • 6.2 Optimization: The smoothed objective can be optimized with Newton’s method, while the stopping criterion accounts for the smoothing-induced duality gap.A gap of ε × min{p, q} for the smoothed problem yields at most (1 + 2 log 2)ε × min{p, q} for the original problem.
  • 6.2 Optimization: The regularization path can be restricted because solutions are exactly zero for λ > ∥Q∥2 and approach the unregularized solution as λ becomes small.The paper proposes logarithmic-grid sampling or predictor-corrector path following over a suitable interval.
  • 6.3 Simulations: In consistent settings, regular and adaptive trace-norm paths jointly estimate singular values and rank, with adaptive methods widening the range of correct rank selection.The simulations use r = 2, p = q = 4 and compare regularization with adaptive exponents γ = 1/2 and γ = 1.
  • 6.3 Simulations: In inconsistent settings, non-adaptive regularization cannot achieve rank and regular consistency together, whereas adaptive schemes can.Across sample sizes, adaptive methods retain high-probability rank selection while estimation error improves; errors also increase as log10(∥Λ∥2) moves above zero.

7. Conclusion

The paper analyzes rank consistency for trace-norm penalization and derives general necessary and sufficient conditions. It identifies extensions to broader losses, collaborative filtering, growing dimensions, and non-asymptotic analysis.

  • The paper derives general necessary and sufficient conditions for rank consistency under trace-norm penalization.
  • Future work includes extending the analysis beyond square loss to classification and other supervised-learning problems.
  • The authors also identify collaborative filtering with partial observations and growing dimensions as extensions of the current setting.
  • Non-asymptotic extensions are proposed through connections with recent work on matrix completion and sparsity consistency.

Appendix A. Tools for analysis of singular value decomposition

The appendix develops singular-value perturbation tools and trace-norm geometry used to analyze rank changes, singular subspaces, and local trace-norm behavior.

  • Singular-value decompositions represent W through positive singular values and associated left and right singular vectors, with rotations required when values coalesce.
  • Spectral projectors isolate invariant subspaces of the symmetric dilation, enabling perturbation analysis of singular subspaces.
  • Under ∥∆∥2 < sr/4, the leading spectral projection admits a first-order perturbation approximation with a bounded spectral-norm remainder.
  • If 4 times the projected perturbation exceeds the smallest positive singular-value threshold, rank(W + ∆) becomes larger than rank(W).
  • The trace norm is dual to the operator norm, and its subdifferential and directional derivatives are expressed using the singular subspaces and orthogonal complements.
  • For perturbations bounded by sr/4, the appendix gives a finer upper bound on trace-norm error beyond the directional-derivative limit.

C.1 Proof of Lemma 2

The proof establishes asymptotic behavior of sampled matrix-prediction quantities using moment assumptions, martingale differences, and M-estimation consistency.

  • Sampling-without-replacement properties establish the required covariance and moment control for the sampled matrix observations.
  • A martingale central limit theorem yields asymptotic normality with covariance σ2Σyy ⊗Σxx.
  • Pointwise convergence of the empirical objective to a uniquely minimized population objective supports consistency through standard M-estimation results.

C.3 Proof of Proposition 7

The proof derives asymptotic normality for the scaled estimation error using epiconvergence, then uses this limit to establish the proposition’s rank behavior.

  • The scaled error ˆ∆ = n^1/2(ˆW − W) is asymptotically normal with mean zero and covariance σ^2Σ−1.
  • The objective functions Vn converge in probability pointwise to a strictly convex limit V, which has a unique global minimum.
  • A simpler M-estimation argument gives ˆW = W + n−1/2ˆ∆ + op(n−1/2), where ˆ∆ is the unique global minimizer of the limiting problem.
  • The covariance structure can be represented using A = n^1/2ˆΣMε, which is asymptotically normal with the correct moments.

C.5 Proof of Proposition 9

The proof characterizes rank consistency through the limiting optimization problem and shows that its probability converges to a limit a strictly between zero and one.

  • The upper-bound argument studies events controlling the orthogonal singular-vector component and shows lim supn→∞P(rank(ˆW) = r) ≤ a.
  • The probability of rank consistency converges to a = P(∥Λ(A)∥2 ⩽ λ0), with a ∈ (0, 1).The limit follows from convergence in distribution of ˆ∆ to the optimizer ∆(A) and the normal distribution of A.
  • The lower-bound argument uses a sufficient condition U⊥⊤∆V⊥ = 0 and establishes lim infn→∞P(rank(ˆW) = r) ≥ a.
  • The constrained optimization condition is equivalent to ∥Λ∥2 ≤ 1, obtained from nonnegativity of the directional derivative.
  • Under λnn^1/2 → ∞ and λn → 0, regular consistency and a positive eigengap control the singular-vector projections, while the remaining terms are asymptotically smaller than λnn^1/2.
  • If accumulation points of n^1/2λn are zero or +∞, Propositions 7 and 9 rule out rank consistency; otherwise a suitable subsequence satisfies the required scaling condition.

C.10 Proof of Theorem 15

The theorem’s proof analyzes the adaptive trace-norm estimator through singular-vector decompositions and epiconvergence, yielding a constrained asymptotically normal limit.

  • The adaptive weighting uses singular-value decay and simultaneous decompositions to compare the leading r singular directions with the remaining columns and singular values.
  • The analysis restricts the relevant perturbations to the rank-r singular-vector subspace when U⊥⊤∆ = 0 and ∆V⊥ = 0.
  • When perturbations have orthogonal components, the trace-norm penalty becomes larger than the corresponding in-subspace expression, with the difference controlled through the weighted singular-value decomposition.
  • The adaptive estimator’s scaled error ˆ∆A converges in distribution to the minimum of a limiting optimization problem because Σmm is invertible.
  • The limiting minimum has the form ∆ = UΘV⊤, where Θ is asymptotically normal with mean zero and covariance determined by Σmm−1.
  • Regular consistency makes the first term of order λnn^1/2, while the other three terms are smaller than Op(n^1/2λn), completing the rank argument.
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