Landscape analysis of an improved power method for tensor decomposition

• URL: http://arxiv.org/abs/2110.15821v1
• Date: Fri, 29 Oct 2021 14:32:54 GMT
• Title: Landscape analysis of an improved power method for tensor decomposition
• Authors: Joe Kileel, Timo Klock, Jo\~ao M. Pereira
• Abstract summary: We consider the optimization formulation for symmetric tensor decomposition recently introduced in the Subspace Power Method. We obtain a near-global guarantee up to $tildeo(Dlfloor m r)$, and a global guarantee up to rank random tensor component.
• Score: 2.363388546004777
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this work, we consider the optimization formulation for symmetric tensor decomposition recently introduced in the Subspace Power Method (SPM) of Kileel and Pereira. Unlike popular alternative functionals for tensor decomposition, the SPM objective function has the desirable properties that its maximal value is known in advance, and its global optima are exactly the rank-1 components of the tensor when the input is sufficiently low-rank. We analyze the non-convex optimization landscape associated with the SPM objective. Our analysis accounts for working with noisy tensors. We derive quantitative bounds such that any second-order critical point with SPM objective value exceeding the bound must equal a tensor component in the noiseless case, and must approximate a tensor component in the noisy case. For decomposing tensors of size $D^{\times m}$, we obtain a near-global guarantee up to rank $\widetilde{o}(D^{\lfloor m/2 \rfloor})$ under a random tensor model, and a global guarantee up to rank $\mathcal{O}(D)$ assuming deterministic frame conditions. This implies that SPM with suitable initialization is a provable, efficient, robust algorithm for low-rank symmetric tensor decomposition. We conclude with numerics that show a practical preferability for using the SPM functional over a more established counterpart.

Related papers

• A Novel Tensor Factorization-Based Method with Robustness to Inaccurate Rank Estimation [9.058215418134209]
  We propose a new tensor norm with a dual low-rank constraint, which utilizes the low-rank prior and rank information at the same time. It is proven theoretically that the resulting tensor completion model can effectively avoid performance degradation caused by inaccurate rank estimation. Based on this, the total cost at each iteration of the optimization algorithm is reduced to $mathcalO(n3log n +kn3)$ from $mathcalO(n4)$ achieved with standard methods.
  arXiv  Detail & Related papers  (2023-05-19T06:26:18Z)
• Tensor Recovery Based on Tensor Equivalent Minimax-Concave Penalty [3.0711362702464675]
  It is an important problem in computer and machine learning. We propose two adaptive models for two tensor recovery problems. The proposed method is superior to state-of-arts experiments.
  arXiv  Detail & Related papers  (2022-01-30T03:28:01Z)
• When Random Tensors meet Random Matrices [50.568841545067144]
  This paper studies asymmetric order-$d$ spiked tensor models with Gaussian noise. We show that the analysis of the considered model boils down to the analysis of an equivalent spiked symmetric textitblock-wise random matrix.
  arXiv  Detail & Related papers  (2021-12-23T04:05:01Z)
• Robust M-estimation-based Tensor Ring Completion: a Half-quadratic Minimization Approach [14.048989759890475]
  We develop a robust approach to tensor ring completion that uses an M-estimator as its error statistic. We present two HQ-based algorithms based on truncated singular value decomposition and matrix factorization.
  arXiv  Detail & Related papers  (2021-06-19T04:37:50Z)
• MTC: Multiresolution Tensor Completion from Partial and Coarse Observations [49.931849672492305]
  Existing completion formulation mostly relies on partial observations from a single tensor. We propose an efficient Multi-resolution Completion model (MTC) to solve the problem.
  arXiv  Detail & Related papers  (2021-06-14T02:20:03Z)
• High Probability Complexity Bounds for Non-Smooth Stochastic Optimization with Heavy-Tailed Noise [51.31435087414348]
  It is essential to theoretically guarantee that algorithms provide small objective residual with high probability. Existing methods for non-smooth convex optimization have complexity bounds with dependence on confidence level. We propose novel stepsize rules for two methods with gradient clipping.
  arXiv  Detail & Related papers  (2021-06-10T17:54:21Z)
• Beyond Lazy Training for Over-parameterized Tensor Decomposition [69.4699995828506]
  We show that gradient descent on over-parametrized objective could go beyond the lazy training regime and utilize certain low-rank structure in the data. Our results show that gradient descent on over-parametrized objective could go beyond the lazy training regime and utilize certain low-rank structure in the data.
  arXiv  Detail & Related papers  (2020-10-22T00:32:12Z)
• Robust Tensor Principal Component Analysis: Exact Recovery via Deterministic Model [5.414544833902815]
  This paper proposes a new method to analyze Robust tensor principal component analysis (RTPCA) It is based on the recently developed tensor-tensor product and tensor singular value decomposition (t-SVD)
  arXiv  Detail & Related papers  (2020-08-05T16:26:10Z)
• Uncertainty quantification for nonconvex tensor completion: Confidence intervals, heteroscedasticity and optimality [92.35257908210316]
  We study the problem of estimating a low-rank tensor given incomplete and corrupted observations. We find that it attains unimprovable rates $ell-2$ accuracy.
  arXiv  Detail & Related papers  (2020-06-15T17:47:13Z)
• Tensor completion via nonconvex tensor ring rank minimization with guaranteed convergence [16.11872681638052]
  In recent studies, the tensor ring (TR) rank has shown high effectiveness in tensor completion. A recently proposed TR rank is based on capturing the structure within the weighted sum penalizing the singular value equally. In this paper, we propose to use the logdet-based function as a non smooth relaxation for solutions practice.
  arXiv  Detail & Related papers  (2020-05-14T03:13:17Z)
• Spectral density estimation with the Gaussian Integral Transform [91.3755431537592]
  spectral density operator $hatrho(omega)=delta(omega-hatH)$ plays a central role in linear response theory. We describe a near optimal quantum algorithm providing an approximation to the spectral density.
  arXiv  Detail & Related papers  (2020-04-10T03:14:38Z)

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.