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We propose a contemporaneous bilinear transformation for matrix time series to alleviate the difficulties in modelling and forecasting large number of time series together. More precisely the transformed matrix splits into several small matrices, and those small matrix series are uncorrelated across all times. Hence an effective dimension-reduction is achieved by modelling each of those small matrix series separately without the loss of information on the overall linear dynamics. We adopt the bilinear transformation such that the rows and the columns of the matrix do not mix together, as they typically represent radically different features. As the targeted transformation is not unique, we identify an ideal version through a new normalization, which facilitates the no-cancellation accumulation of the information from different time lags. The non-asymptotic error bounds of the estimated transformation are derived, leading to the uniform convergence rates of the estimation. The proposed method is illustrated numerically via both simulated and real data examples.

This paper provides a new abstract stability result for perturbed saddle-point problems which is based on a proper norm fitting. We derive the stability condition according to Babu\v{s}ka's theory from a small inf-sup condition, similar to the famous Ladyzhenskaya-Babu\v{s}ka-Brezzi (LBB) condition, and the other standard assumptions in Brezzi's theory under the resulting combined norm. The proposed framework allows to split the norms into proper seminorms and not only results in simpler (shorter) proofs of many stability results but also guides the construction of parameter robust norm-equivalent preconditioners. These benefits are demonstrated with several examples arising from different formulations of Biot's model of consolidation.

In this work, we investigate the universal representation capacity of the Matrix Product States (MPS) from the perspective of boolean functions and continuous functions. We show that MPS can accurately realize arbitrary boolean functions by providing a construction method of the corresponding MPS structure for an arbitrarily given boolean gate. Moreover, we prove that the function space of MPS with the scale-invariant sigmoidal activation is dense in the space of continuous functions defined on a compact subspace of the $n$-dimensional real coordinate space $\mathbb{R^{n}}$. We study the relation between MPS and neural networks and show that the MPS with a scale-invariant sigmoidal function is equivalent to a one-hidden-layer neural network equipped with a kernel function. We construct the equivalent neural networks for several specific MPS models and show that non-linear kernels such as the polynomial kernel which introduces the couplings between different components of the input into the model appear naturally in the equivalent neural networks. At last, we discuss the realization of the Gaussian Process (GP) with infinitely wide MPS by studying their equivalent neural networks.

This paper studies bilinear saddle point problems $\min_{\bf{x}} \max_{\bf{y}} g(\bf{x}) + \bf{x}^{\top} \bf{A} \bf{y} - h(\bf{y})$, where the functions $g, h$ are smooth and strongly-convex. When the gradient and proximal oracle related to $g$ and $h$ are accessible, optimal algorithms have already been developed in the literature \cite{chambolle2011first, palaniappan2016stochastic}. However, the proximal operator is not always easy to compute, especially in constraint zero-sum matrix games \cite{zhang2020sparsified}. This work proposes a new algorithm which only requires the access to the gradients of $g, h$. Our algorithm achieves a complexity upper bound $\tilde{\mathcal{O}}\left( \frac{\|\bf{A}\|_2}{\sqrt{\mu_x \mu_y}} + \sqrt[4]{\kappa_x \kappa_y (\kappa_x + \kappa_y)} \right)$ which has optimal dependency on the coupling condition number $\frac{\|\bf{A}\|_2}{\sqrt{\mu_x \mu_y}}$ up to logarithmic factors.

We investigate the asymptotic risk of a general class of overparameterized likelihood models, including deep models. The recent empirical success of large-scale models has motivated several theoretical studies to investigate a scenario wherein both the number of samples, $n$, and parameters, $p$, diverge to infinity and derive an asymptotic risk at the limit. However, these theorems are only valid for linear-in-feature models, such as generalized linear regression, kernel regression, and shallow neural networks. Hence, it is difficult to investigate a wider class of nonlinear models, including deep neural networks with three or more layers. In this study, we consider a likelihood maximization problem without the model constraints and analyze the upper bound of an asymptotic risk of an estimator with penalization. Technically, we combine a property of the Fisher information matrix with an extended Marchenko-Pastur law and associate the combination with empirical process techniques. The derived bound is general, as it describes both the double descent and the regularized risk curves, depending on the penalization. Our results are valid without the linear-in-feature constraints on models and allow us to derive the general spectral distributions of a Fisher information matrix from the likelihood. We demonstrate that several explicit models, such as parallel deep neural networks, ensemble learning, and residual networks, are in agreement with our theory. This result indicates that even large and deep models have a small asymptotic risk if they exhibit a specific structure, such as divisibility. To verify this finding, we conduct a real-data experiment with parallel deep neural networks. Our results expand the applicability of the asymptotic risk analysis, and may also contribute to the understanding and application of deep learning.

Consider the problem of finding the maximal nonpositive solvent $\Phi$ of the quadratic matrix equation (QME) $X^2 + BX + C =0$ with $B$ being a nonsingular $M$-matrix and $C$ an $M$-matrix such that $B^{-1}C\ge 0$, and $B - C - I$ a nonsingular $M$-matrix. Such QME arises from an overdamped vibrating system. Recently, Yu et al. ({\em Appl. Math. Comput.}, 218: 3303--3310, 2011) proved that $\rho(\Phi)\le 1$ for this QME. In this paper, we slightly improve their result and prove $\rho(\Phi)< 1$, which is important for the quadratic convergence of the structure-preserving doubling algorithm. Then, a new globally monotonically and quadratically convergent structure-preserving doubling algorithm to solve the QME is developed. Numerical examples are presented to demonstrate the feasibility and effectiveness of our method.

Proximal Policy Optimization (PPO) is a highly popular model-free reinforcement learning (RL) approach. However, in continuous state and actions spaces and a Gaussian policy -- common in computer animation and robotics -- PPO is prone to getting stuck in local optima. In this paper, we observe a tendency of PPO to prematurely shrink the exploration variance, which naturally leads to slow progress. Motivated by this, we borrow ideas from CMA-ES, a black-box optimization method designed for intelligent adaptive Gaussian exploration, to derive PPO-CMA, a novel proximal policy optimization approach that can expand the exploration variance on objective function slopes and shrink the variance when close to the optimum. This is implemented by using separate neural networks for policy mean and variance and training the mean and variance in separate passes. Our experiments demonstrate a clear improvement over vanilla PPO in many difficult OpenAI Gym MuJoCo tasks.

We show that for the problem of testing if a matrix $A \in F^{n \times n}$ has rank at most $d$, or requires changing an $\epsilon$-fraction of entries to have rank at most $d$, there is a non-adaptive query algorithm making $\widetilde{O}(d^2/\epsilon)$ queries. Our algorithm works for any field $F$. This improves upon the previous $O(d^2/\epsilon^2)$ bound (SODA'03), and bypasses an $\Omega(d^2/\epsilon^2)$ lower bound of (KDD'14) which holds if the algorithm is required to read a submatrix. Our algorithm is the first such algorithm which does not read a submatrix, and instead reads a carefully selected non-adaptive pattern of entries in rows and columns of $A$. We complement our algorithm with a matching query complexity lower bound for non-adaptive testers over any field. We also give tight bounds of $\widetilde{\Theta}(d^2)$ queries in the sensing model for which query access comes in the form of $\langle X_i, A\rangle:=tr(X_i^\top A)$; perhaps surprisingly these bounds do not depend on $\epsilon$. We next develop a novel property testing framework for testing numerical properties of a real-valued matrix $A$ more generally, which includes the stable rank, Schatten-$p$ norms, and SVD entropy. Specifically, we propose a bounded entry model, where $A$ is required to have entries bounded by $1$ in absolute value. We give upper and lower bounds for a wide range of problems in this model, and discuss connections to the sensing model above.

Learning embedding functions, which map semantically related inputs to nearby locations in a feature space supports a variety of classification and information retrieval tasks. In this work, we propose a novel, generalizable and fast method to define a family of embedding functions that can be used as an ensemble to give improved results. Each embedding function is learned by randomly bagging the training labels into small subsets. We show experimentally that these embedding ensembles create effective embedding functions. The ensemble output defines a metric space that improves state of the art performance for image retrieval on CUB-200-2011, Cars-196, In-Shop Clothes Retrieval and VehicleID.

We propose a new method of estimation in topic models, that is not a variation on the existing simplex finding algorithms, and that estimates the number of topics K from the observed data. We derive new finite sample minimax lower bounds for the estimation of A, as well as new upper bounds for our proposed estimator. We describe the scenarios where our estimator is minimax adaptive. Our finite sample analysis is valid for any number of documents (n), individual document length (N_i), dictionary size (p) and number of topics (K), and both p and K are allowed to increase with n, a situation not handled well by previous analyses. We complement our theoretical results with a detailed simulation study. We illustrate that the new algorithm is faster and more accurate than the current ones, although we start out with a computational and theoretical disadvantage of not knowing the correct number of topics K, while we provide the competing methods with the correct value in our simulations.