Parameter Estimation for Generalized Low-Rank Matrix Sensing by Learning on Riemannian Manifolds

• URL: http://arxiv.org/abs/2407.10238v1
• Date: Sun, 14 Jul 2024 15:11:13 GMT
• Title: Parameter Estimation for Generalized Low-Rank Matrix Sensing by Learning on Riemannian Manifolds
• Authors: Osbert Bastani,
• Abstract summary: We prove convergence guarantees for generalized low-rank matrix sensing. We focus on local convergence of the optimal estimator, ignoring questions of optimization. Our analysis relies on tools from Riemannian geometry to handle the rotational symmetry in the parameter space.
• Score: 37.53442095760427
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We prove convergence guarantees for generalized low-rank matrix sensing -- i.e., where matrix sensing where the observations may be passed through some nonlinear link function. We focus on local convergence of the optimal estimator, ignoring questions of optimization. In particular, assuming the minimizer of the empirical loss $\theta^0$ is in a constant size ball around the true parameters $\theta^*$, we prove that $d(\theta^0,\theta^*)=\tilde{O}(\sqrt{dk^2/n})$. Our analysis relies on tools from Riemannian geometry to handle the rotational symmetry in the parameter space.

Related papers

• The Communication Complexity of Approximating Matrix Rank [50.6867896228563]
  We show that this problem has randomized communication complexity $Omega(frac1kcdot n2log|mathbbF|)$. As an application, we obtain an $Omega(frac1kcdot n2log|mathbbF|)$ space lower bound for any streaming algorithm with $k$ passes.
  arXiv  Detail & Related papers  (2024-10-26T06:21:42Z)
• Provably learning a multi-head attention layer [55.2904547651831]
  Multi-head attention layer is one of the key components of the transformer architecture that sets it apart from traditional feed-forward models. In this work, we initiate the study of provably learning a multi-head attention layer from random examples. We prove computational lower bounds showing that in the worst case, exponential dependence on $m$ is unavoidable.
  arXiv  Detail & Related papers  (2024-02-06T15:39:09Z)
• Universality for the global spectrum of random inner-product kernel matrices in the polynomial regime [12.221087476416056]
  In this paper, we show that this phenomenon is universal, holding as soon as $X$ has i.i.d. entries with all finite moments. In the case of non-integer $ell$, the Marvcenko-Pastur term disappears.
  arXiv  Detail & Related papers  (2023-10-27T17:15:55Z)
• $O(k)$-Equivariant Dimensionality Reduction on Stiefel Manifolds [2.0818404738530525]
  Many real-world datasets live on high-dimensional Stiefel and Grassmannian manifold, $V_k(mathbbRN)$ and $Gr(k, mathbbRN)$ respectively. We propose an algorithm called textitPrincipal Stiefel Coordinates (PSC) to reduce data dimensionality from $ V_k(mathbbRN)$ to $V_k(mathbbRn)$ in an textit$O(k)$-equivariant manner
  arXiv  Detail & Related papers  (2023-09-19T17:21:12Z)
• A General Algorithm for Solving Rank-one Matrix Sensing [15.543065204102714]
  The goal of matrix sensing is to recover a matrix $A_star in mathbbRn times n$, based on a sequence of measurements. In this paper, we relax that rank-$k$ assumption and solve a much more general matrix sensing problem.
  arXiv  Detail & Related papers  (2023-03-22T04:07:26Z)
• Spectral properties of sample covariance matrices arising from random matrices with independent non identically distributed columns [50.053491972003656]
  It was previously shown that the functionals $texttr(AR(z))$, for $R(z) = (frac1nXXT- zI_p)-1$ and $Ain mathcal M_p$ deterministic, have a standard deviation of order $O(|A|_* / sqrt n)$. Here, we show that $|mathbb E[R(z)] - tilde R(z)|_F
  arXiv  Detail & Related papers  (2021-09-06T14:21:43Z)
• On the computational and statistical complexity of over-parameterized matrix sensing [30.785670369640872]
  We consider solving the low rank matrix sensing problem with Factorized Gradient Descend (FGD) method. By decomposing the factorized matrix $mathbfF$ into separate column spaces, we show that $|mathbfF_t - mathbfF_t - mathbfX*|_F2$ converges to a statistical error.
  arXiv  Detail & Related papers  (2021-01-27T04:23:49Z)
• Sparse sketches with small inversion bias [79.77110958547695]
  Inversion bias arises when averaging estimates of quantities that depend on the inverse covariance. We develop a framework for analyzing inversion bias, based on our proposed concept of an $(epsilon,delta)$-unbiased estimator for random matrices. We show that when the sketching matrix $S$ is dense and has i.i.d. sub-gaussian entries, the estimator $(epsilon,delta)$-unbiased for $(Atop A)-1$ with a sketch of size $m=O(d+sqrt d/
  arXiv  Detail & Related papers  (2020-11-21T01:33:15Z)
• The Average-Case Time Complexity of Certifying the Restricted Isometry Property [66.65353643599899]
  In compressed sensing, the restricted isometry property (RIP) on $M times N$ sensing matrices guarantees efficient reconstruction of sparse vectors. We investigate the exact average-case time complexity of certifying the RIP property for $Mtimes N$ matrices with i.i.d. $mathcalN(0,1/M)$ entries.
  arXiv  Detail & Related papers  (2020-05-22T16:55:01Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.