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The singular value decomposition (SVD) allows to put a matrix as a product of three matrices: a matrix with the left singular vectors, a matrix with the positive-valued singular values and a matrix with the right singular vectors. There are two main approaches allowing to get the SVD result: the classical method and the randomized method. The analysis of the classical approach leads to accurate singular values. The randomized approach is especially used for high dimensional matrix and is based on the approximation accuracy without computing necessary all singular values. In this paper, the SVD computation is formalized as an optimization problem and a use of the gradient search algorithm. That results in a power method allowing to get all or the first largest singular values and their associated right vectors. In this iterative search, the accuracy on the singular values and the associated vector matrix depends on the user settings. Two applications of the SVD are the principal component analysis and the autoencoder used in the neural network models.

Counting logics with a bounded number of variables form one of the central concepts in descriptive complexity theory. Although they restrict the number of variables that a formula can contain, the variables can be nested within scopes of quantified occurrences of themselves. In other words, the variables can be requantified. We study the fragments obtained from counting logics by restricting requantification for some but not necessarily all the variables. Similar to the logics without limitation on requantification, we develop tools to investigate the restricted variants. Specifically, we introduce a bijective pebble game in which certain pebbles can only be placed once and for all, and a corresponding two-parametric family of Weisfeiler-Leman algorithms. We show close correspondences between the three concepts. By using a suitable cops-and-robber game and adaptations of the Cai-F\"urer-Immerman construction, we completely clarify the relative expressive power of the new logics. We show that the restriction of requantification has beneficial algorithmic implications in terms of graph identification. Indeed, we argue that with regard to space complexity, non-requantifiable variables only incur an additive polynomial factor when testing for equivalence. In contrast, for all we know, requantifiable variables incur a multiplicative linear factor. Finally, we observe that graphs of bounded tree-depth and 3-connected planar graphs can be identified using no, respectively, only a very limited number of requantifiable variables.

We consider the approximation of functions by 2-layer neural networks with a small number of hidden weights based on the squared loss and small datasets. Due to the highly non-convex energy landscape, gradient-based training often suffers from local minima. As a remedy, we initialize the hidden weights with samples from a learned proposal distribution, which we parameterize as a deep generative model. To train this model, we exploit the fact that with fixed hidden weights, the optimal output weights solve a linear equation. After learning the generative model, we refine the sampled weights with a gradient-based post-processing in the latent space. Here, we also include a regularization scheme to counteract potential noise. Finally, we demonstrate the effectiveness of our approach by numerical examples.

Classical simulations of quantum circuits are essential for verifying and benchmarking quantum algorithms, particularly for large circuits, where computational demands increase exponentially with the number of qubits. Among available methods, the classical simulation of quantum circuits inspired by density functional theory -- the so-called QC-DFT method, shows promise for large circuit simulations as it approximates the quantum circuits using single-qubit reduced density matrices to model multi-qubit systems. However, the QC-DFT method performs very poorly when dealing with multi-qubit gates. In this work, we introduce a novel CNOT "functional" that leverages neural networks to generate unitary transformations, effectively mitigating the simulation errors observed in the original QC-DFT method. For random circuit simulations, our modified QC-DFT enables efficient computation of single-qubit marginal measurement probabilities, or single-qubit probability (SQPs), and achieves lower SQP errors and higher fidelities than the original QC-DFT method. Despite limitations in capturing full entanglement and joint probability distributions, we find potential applications of SQPs in simulating Shor's and Grover's algorithms for specific solution classes. These findings advance the capabilities of classical simulations for some quantum problems and provide insights into managing entanglement and gate errors in practical quantum computing.

Out-of-distribution generalization capabilities of sequence-to-sequence models can be studied from the lens of two crucial forms of generalization: length generalization -- the ability to generalize to longer sequences than ones seen during training, and compositional generalization: the ability to generalize to token combinations not seen during training. In this work, we provide first provable guarantees on length and compositional generalization for common sequence-to-sequence models -- deep sets, transformers, state space models, and recurrent neural nets -- trained to minimize the prediction error. Taking a first principles perspective, we study the realizable case, i.e., the labeling function is realizable on the architecture. We show that \emph{simple limited capacity} versions of these different architectures achieve both length and compositional generalization. In all our results across different architectures, we find that the learned representations are linearly related to the representations generated by the true labeling function.

Portfolio optimization is a ubiquitous problem in financial mathematics that relies on accurate estimates of covariance matrices for asset returns. However, estimates of pairwise covariance could be better and calculating time-sensitive optimal portfolios is energy-intensive for digital computers. We present an energy-efficient, fast, and fully analog pipeline for solving portfolio optimization problems that overcomes these limitations. The analog paradigm leverages the fundamental principles of physics to recover accurate optimal portfolios in a two-step process. Firstly, we utilize equilibrium propagation, an analog alternative to backpropagation, to train linear autoencoder neural networks to calculate low-rank covariance matrices. Then, analog continuous Hopfield networks output the minimum variance portfolio for a given desired expected return. The entire efficient frontier may then be recovered, and an optimal portfolio selected based on risk appetite.

We propose a scalable preconditioned primal-dual hybrid gradient algorithm for solving partial differential equations (PDEs). We multiply the PDE with a dual test function to obtain an inf-sup problem whose loss functional involves lower-order differential operators. The Primal-Dual Hybrid Gradient (PDHG) algorithm is then leveraged for this saddle point problem. By introducing suitable precondition operators to the proximal steps in the PDHG algorithm, we obtain an alternative natural gradient ascent-descent optimization scheme for updating the neural network parameters. We apply the Krylov subspace method (MINRES) to evaluate the natural gradients efficiently. Such treatment readily handles the inversion of precondition matrices via matrix-vector multiplication. A posterior convergence analysis is established for the time-continuous version of the proposed method. The algorithm is tested on various types of PDEs with dimensions ranging from $1$ to $50$, including linear and nonlinear elliptic equations, reaction-diffusion equations, and Monge-Amp\`ere equations stemming from the $L^2$ optimal transport problems. We compare the performance of the proposed method with several commonly used deep learning algorithms such as physics-informed neural networks (PINNs), the DeepRitz method, weak adversarial networks (WANs), etc, for solving PDEs using the Adam and L-BFGS optimizers. The numerical results suggest that the proposed method performs efficiently and robustly and converges more stably.

This research presents FDASynthesis, a novel algorithm designed to generate synthetic GPS trajectory data while preserving privacy. After pre-processing the input GPS data, human mobility traces are modeled as multidimensional curves using Functional Data Analysis (FDA). Then, the synthesis process identifies the K-nearest trajectories and averages their Square-Root Velocity Functions (SRVFs) to generate synthetic data. This results in synthetic trajectories that maintain the utility of the original data while ensuring privacy. Although applied for human mobility research, FDASynthesis is highly adaptable to different types of functional data, offering a scalable solution in various application domains.

Accurately modeling the correlation structure of errors is critical for reliable uncertainty quantification in probabilistic time series forecasting. While recent deep learning models for multivariate time series have developed efficient parameterizations for time-varying contemporaneous covariance, but they often assume temporal independence of errors for simplicity. However, real-world data often exhibit significant error autocorrelation and cross-lag correlation due to factors such as missing covariates. In this paper, we introduce a plug-and-play method that learns the covariance structure of errors over multiple steps for autoregressive models with Gaussian-distributed errors. To ensure scalable inference and computational efficiency, we model the contemporaneous covariance using a low-rank-plus-diagonal parameterization and capture cross-covariance through a group of independent latent temporal processes. The learned covariance matrix is then used to calibrate predictions based on observed residuals. We evaluate our method on probabilistic models built on RNNs and Transformer architectures, and the results confirm the effectiveness of our approach in improving predictive accuracy and uncertainty quantification without significantly increasing the parameter size.

Multi-relation Question Answering is a challenging task, due to the requirement of elaborated analysis on questions and reasoning over multiple fact triples in knowledge base. In this paper, we present a novel model called Interpretable Reasoning Network that employs an interpretable, hop-by-hop reasoning process for question answering. The model dynamically decides which part of an input question should be analyzed at each hop; predicts a relation that corresponds to the current parsed results; utilizes the predicted relation to update the question representation and the state of the reasoning process; and then drives the next-hop reasoning. Experiments show that our model yields state-of-the-art results on two datasets. More interestingly, the model can offer traceable and observable intermediate predictions for reasoning analysis and failure diagnosis.