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<p>Low-rank matrix recovery is well-known to exhibit <i>benign nonconvexity</i> under the restricted isometry property (RIP): every second-order critical point is globally optimal, so local methods provably recover the ground truth. Motivated by the strong empirical performance of projected gradient methods for nonnegative low-rank recovery problems, we investigate whether this benign geometry persists when the factor matrices are constrained to be elementwise nonnegative. In the simple setting of a rank-1 nonnegative ground truth, we confirm that benign nonconvexity holds in the fully-observed case with RIP constant <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\delta =0\)</EquationSource> </InlineEquation>. This benign nonconvexity, however, is unstable. It fails to extend to the partially-observed case with any arbitrarily small RIP constant <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\delta &gt;0\)</EquationSource> </InlineEquation>, and to higher-rank ground truths <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(r^{\star }&gt;1\)</EquationSource> </InlineEquation>, regardless of how much the search rank <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(r\ge r^{\star }\)</EquationSource> </InlineEquation> is overparameterized. Together, these results undermine the standard stability-based explanation for the empirical success of nonconvex methods and suggest that fundamentally different tools are needed to analyze nonnegative low-rank recovery. </p>

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Nonnegative low-rank matrix recovery can have spurious local minima

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  • 发表日期：

    2026年3月23日

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  • Richard Y. Zhang

摘要

Low-rank matrix recovery is well-known to exhibit benign nonconvexity under the restricted isometry property (RIP): every second-order critical point is globally optimal, so local methods provably recover the ground truth. Motivated by the strong empirical performance of projected gradient methods for nonnegative low-rank recovery problems, we investigate whether this benign geometry persists when the factor matrices are constrained to be elementwise nonnegative. In the simple setting of a rank-1 nonnegative ground truth, we confirm that benign nonconvexity holds in the fully-observed case with RIP constant \(\delta =0\) . This benign nonconvexity, however, is unstable. It fails to extend to the partially-observed case with any arbitrarily small RIP constant \(\delta >0\) , and to higher-rank ground truths \(r^{\star }>1\) , regardless of how much the search rank \(r\ge r^{\star }\) is overparameterized. Together, these results undermine the standard stability-based explanation for the empirical success of nonconvex methods and suggest that fundamentally different tools are needed to analyze nonnegative low-rank recovery.