Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userConvex Tensor Decomposition via Structured Schatten Norm Regularization
547
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We discuss structured Schatten norms for tensor decomposition that includes two recently proposed norms (overlapped and latent) for convex-optimization-based tensor decomposition, and connect tensor decomposition with wider literature on structured sparsity. Based on the properties of the structured Schatten norms, we mathematically analyze the performance of latent approach for tensor decomposition, which was empirically found to perform better than the overlapped approach in some settings. We show theoretically that this is indeed the case. In particular, when the unknown true tensor is low-rank in a specific mode, this approach performs as good as knowing the mode with the smallest rank. Along the way, we show a novel duality result for structures Schatten norms, establish the consistency, and discuss the identifiability of this approach. We confirm through numerical simulations that our theoretical prediction can precisely predict the scaling behavior of the mean squared error.
rate research
Read More
The nuclear norm and Schatten-$p$ quasi-norm of a matrix are popular rank proxies in low-rank matrix recovery. Unfortunately, computing the nuclear norm or Schatten-$p$ quasi-norm of a tensor is NP-hard, which is a pity for low-rank tensor completion (LRTC) and tensor robust principal component analysis (TRPCA). In this paper, we propose a new class of rank regularizers based on the Euclidean norms of the CP component vectors of a tensor and show that these regularizers are monotonic transformations of tensor Schatten-$p$ quasi-norm. This connection enables us to minimize the Schatten-$p$ quasi-norm in LRTC and TRPCA implicitly. The methods do not use the singular value decomposition and hence scale to big tensors. Moreover, the methods are not sensitive to the choice of initial rank and provide an arbitrarily sharper rank proxy for low-rank tensor recovery compared to nuclear norm. We provide theoretical guarantees in terms of recovery error for LRTC and TRPCA, which show relatively smaller $p$ of Schatten-$p$ quasi-norm leads to tighter error bounds. Experiments using LRTC and TRPCA on synthetic data and natural images verify the effectiveness and superiority of our methods compared to baseline methods.
The transition kernel of a continuous-state-action Markov decision process (MDP) admits a natural tensor structure. This paper proposes a tensor-inspired unsupervised learning method to identify meaningful low-dimensional state and action representations from empirical trajectories. The method exploits the MDPs tensor structure by kernelization, importance sampling and low-Tucker-rank approximation. This method can be further used to cluster states and actions respectively and find the best discrete MDP abstraction. We provide sharp statistical error bounds for tensor concentration and the preservation of diffusion distance after embedding.
In this paper, we develop a novel procedure for low-rank tensor regression, namely emph{underline{I}mportance underline{S}ketching underline{L}ow-rank underline{E}stimation for underline{T}ensors} (ISLET). The central idea behind ISLET is emph{importance sketching}, i.e., carefully designed sketches based on both the responses and low-dimensional structure of the parameter of interest. We show that the proposed method is sharply minimax optimal in terms of the mean-squared error under low-rank Tucker assumptions and under randomized Gaussian ensemble design. In addition, if a tensor is low-rank with group sparsity, our procedure also achieves minimax optimality. Further, we show through numerical study that ISLET achieves comparable or better mean-squared error performance to existing state-of-the-art methods while having substantial storage and run-time advantages including capabilities for parallel and distributed computing. In particular, our procedure performs reliable estimation with tensors of dimension $p = O(10^8)$ and is $1$ or $2$ orders of magnitude faster than baseline methods.
Tensor decomposition methods allow us to learn the parameters of latent variable models through decomposition of low-order moments of data. A significant limitation of these algorithms is that there exists no general method to regularize them, and in the past regularization has mostly been performed using bespoke modifications to the algorithms, tailored for the particular form of the desired regularizer. We present a general method of regularizing tensor decomposition methods which can be used for any likelihood model that is learnable using tensor decomposition methods and any differentiable regularization function by supplementing the training data with pseudo-data. The pseudo-data is optimized to balance two terms: being as close as possible to the true data and enforcing the desired regularization. On synthetic, semi-synthetic and real data, we demonstrate that our method can improve inference accuracy and regularize for a broad range of goals including transfer learning, sparsity, interpretability, and orthogonality of the learned parameters.
We consider the problem of recovering a low-rank tensor from its noisy observation. Previous work has shown a recovery guarantee with signal to noise ratio $O(n^{lceil K/2 rceil /2})$ for recovering a $K$th order rank one tensor of size $ntimes cdots times n$ by recursive unfolding. In this paper, we first improve this bound to $O(n^{K/4})$ by a much simpler approach, but with a more careful analysis. Then we propose a new norm called the subspace norm, which is based on the Kronecker products of factors obtained by the proposed simple estimator. The imposed Kronecker structure allows us to show a nearly ideal $O(sqrt{n}+sqrt{H^{K-1}})$ bound, in which the parameter $H$ controls the blend from the non-convex estimator to mode-wise nuclear norm minimization. Furthermore, we empirically demonstrate that the subspace norm achieves the nearly ideal denoising performance even with $H=O(1)$.
suggested questions
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
