Related papers: Average-Case Reductions for $k$-XOR and Tensor PCA
Principal component analysis (PCA) is a classical method for dimensionality reduction based on extracting the dominant eigenvectors of the sample covariance matrix. However, PCA is well known to behave poorly in the ``large $p$, small $n$''…
Recovering a low-rank tensor from incomplete information is a recurring problem in signal processing and machine learning. The most popular convex relaxation of this problem minimizes the sum of the nuclear norms of the unfoldings of the…
Tensor computations, with matrix multiplication being the primary operation, serve as the fundamental basis for data analysis, physics, machine learning, and deep learning. As the scale and complexity of data continue to grow rapidly, the…
We study the problem of designing worst-case to average-case reductions for quantum algorithms. For all linear problems, we provide an explicit and efficient transformation of quantum algorithms that are only correct on a small (even…
In this paper, we study the estimation of a rank-one spiked tensor in the presence of heavy tailed noise. Our results highlight some of the fundamental similarities and differences in the tradeoff between statistical and computational…
Many applications in data science and scientific computing involve large-scale datasets that are expensive to store and compute with, but can be efficiently compressed and stored in an appropriate tensor format. In recent years, randomized…
We consider the problem of learning a linear factor model. We propose a regularized form of principal component analysis (PCA) and demonstrate through experiments with synthetic and real data the superiority of resulting estimates to those…
Tensor networks are powerful factorization techniques which reduce resource requirements for numerically simulating principal quantum many-body systems and algorithms. The computational complexity of a tensor network simulation depends on…
Principal Component Analysis (PCA) is the most widely used tool for linear dimensionality reduction and clustering. Still it is highly sensitive to outliers and does not scale well with respect to the number of data samples. Robust PCA…
Many machine learning systems are vulnerable to small perturbations made to inputs either at test time or at training time. This has received much recent interest on the empirical front due to applications where reliability and security are…
Canonical Polyadic Decomposition (CPD) of a third-order tensor is a minimal decomposition into a sum of rank-$1$ tensors. We find new mild deterministic conditions for the uniqueness of individual rank-$1$ tensors in CPD and present an…
Principal component analysis (PCA) is a dimensionality reduction method in data analysis that involves diagonalizing the covariance matrix of the dataset. Recently, quantum algorithms have been formulated for PCA based on diagonalizing a…
In the context of sparse principal component detection, we bring evidence towards the existence of a statistical price to pay for computational efficiency. We measure the performance of a test by the smallest signal strength that it can…
Many applications, such as photon-limited imaging and genomics, involve large datasets with noisy entries from exponential family distributions. It is of interest to estimate the covariance structure and principal components of the…
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…
The low-rank matrix approximation problem with respect to the component-wise $\ell_1$-norm ($\ell_1$-LRA), which is closely related to robust principal component analysis (PCA), has become a very popular tool in data mining and machine…
Spectral methods have been the mainstay in several domains such as machine learning and scientific computing. They involve finding a certain kind of spectral decomposition to obtain basis functions that can capture important structures for…
In this paper, the averaging principle is studied for a class of multiscale stochastic partial differential equations driven by $\alpha$-stable process, where $\alpha\in(1,2)$. Using the technique of Poisson equation, the orders of strong…
We derive and implement a new way of solving coupled cluster equations with lower computational scaling. Our method is based on decomposition of both amplitudes and two electron integrals, using a combination of tensor hypercontraction and…
Low rank tensor approximation is a fundamental tool in modern machine learning and data science. In this paper, we study the characterization, perturbation analysis, and an efficient sampling strategy for two primary tensor CUR…