相关论文: Generalized rank-constrained matrix approximations

In this paper, we study the problem of approximately computing the product of two real matrices. In particular, we analyze a dimensionality-reduction-based approximation algorithm due to Sarlos [1], introducing the notion of nuclear rank as…

Mathematically characterizing the implicit regularization induced by gradient-based optimization is a longstanding pursuit in the theory of deep learning. A widespread hope is that a characterization based on minimization of norms may…

In this paper we announce a conjecture concerning enumeration of 2n x k n-times persymmetric matrices over F_2 by rank.

Low rank regularization, in essence, involves introducing a low rank or approximately low rank assumption for matrix we aim to learn, which has achieved great success in many fields including machine learning, data mining and computer…

In this paper, we show that the low rank matrix completion problem can be reduced to the problem of finding the rank of a certain tensor.

The Frobenius method can be used to compute solutions of ordinary linear differential equations by generalized power series. Each series converges in a circle which at least extends to the nearest singular point; hence exponentially fast…

Given a matrix-valued function $\mathcal{F}(\lambda)=\sum_{i=1}^d f_i(\lambda) A_i$, with complex matrices $A_i$ and $f_i(\lambda)$ entire functions for $i=1,\ldots,d$, we discuss a method for the numerical approximation of the distance to…

The central problem in this work is to compute a ranking of a set of elements which is "closest to" a given set of input rankings of the elements. We define "closest to" in an established way as having the minimum sum of Kendall-Tau…

In this note, we investigate how well we can reconstruct the best rank-$r$ approximation of a large matrix from a small number of its entries. We show that even if a data matrix is of full rank and cannot be approximated well by a low-rank…

1. A standard Gaussian random matrix has full rank with probability 1 and is well-conditioned with a probability quite close to 1 and converging to 1 fast as the matrix deviates from square shape and becomes more rectangular. 2. If we…

We consider multilevel low rank (MLR) matrices, defined as a row and column permutation of a sum of matrices, each one a block diagonal refinement of the previous one, with all blocks low rank given in factored form. MLR matrices extend low…

The aim of reduced rank regression is to connect multiple response variables to multiple predictors. This model is very popular, especially in biostatistics where multiple measurements on individuals can be re-used to predict multiple…

Rank deficient Hankel matrices are at the core of several applications. However, in practice, the coefficients of these matrices are noisy due to e.g. measurements errors and computational errors, so generically the involved matrices are…

The fundamental matrix can be estimated from point matches. The current gold standard is to bootstrap the eight-point algorithm and two-view projective bundle adjustment. The eight-point algorithm first computes a simple linear least…

We introduce a "learning-based" algorithm for the low-rank decomposition problem: given an $n \times d$ matrix $A$, and a parameter $k$, compute a rank-$k$ matrix $A'$ that minimizes the approximation loss $\|A-A'\|_F$. The algorithm uses a…

We study the maximal rank in affine subspaces of symmetric or alternating matrices, in terms of the matching numbers of certain associated graphs. Applications include simple proofs of upper bounds on the dimension of such subspaces in…

Randomized algorithms are overwhelming methods for low-rank approximation that can alleviate the computational expenditure with great reliability compared to deterministic algorithms. A crucial thought is generating a standard Gaussian…

Low rank approximation is a commonly occurring problem in many computer vision and machine learning applications. There are two common ways of optimizing the resulting models. Either the set of matrices with a given rank can be explicitly…

Recht, Fazel, and Parrilo provided an analogy between rank minimization and $\ell_0$-norm minimization. Subject to the rank-restricted isometry property, nuclear norm minimization is a guaranteed algorithm for rank minimization. The…

We consider robust low rank matrix estimation as a trace regression when outputs are contaminated by adversaries. The adversaries are allowed to add arbitrary values to arbitrary outputs. Such values can depend on any samples. We deal with…