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We present a new method of self-supervised learning and knowledge distillation based on the multi-views and multi-representations (MV-MR). The MV-MR is based on the maximization of dependence between learnable embeddings from augmented and non-augmented views, jointly with the maximization of dependence between learnable embeddings from augmented view and multiple non-learnable representations from non-augmented view. We show that the proposed method can be used for efficient self-supervised classification and model-agnostic knowledge distillation. Unlike other self-supervised techniques, our approach does not use any contrastive learning, clustering, or stop gradients. MV-MR is a generic framework allowing the incorporation of constraints on the learnable embeddings via the usage of image multi-representations as regularizers. Along this line, knowledge distillation is considered a particular case of such a regularization. MV-MR provides the state-of-the-art performance on the STL10 and ImageNet-1K datasets among non-contrastive and clustering-free methods. We show that a lower complexity ResNet50 model pretrained using proposed knowledge distillation based on the CLIP ViT model achieves state-of-the-art performance on STL10 linear evaluation. The code is available at: //github.com/vkinakh/mv-mr

We study an interacting particle method (IPM) for computing the large deviation rate function of entropy production for diffusion processes, with emphasis on the vanishing-noise limit and high dimensions. The crucial ingredient to obtain the rate function is the computation of the principal eigenvalue $\lambda$ of elliptic, non-self-adjoint operators. We show that this principal eigenvalue can be approximated in terms of the spectral radius of a discretized evolution operator obtained from an operator splitting scheme and an Euler--Maruyama scheme with a small time step size, and we show that this spectral radius can be accessed through a large number of iterations of this discretized semigroup, suitable for the IPM. The IPM applies naturally to problems in unbounded domains, scales easily to high dimensions, and adapts to singular behaviors in the vanishing-noise limit. We show numerical examples in dimensions up to 16. The numerical results show that our numerical approximation of $\lambda$ converges to the analytical vanishing-noise limit within visual tolerance with a fixed number of particles and a fixed time step size. Our paper appears to be the first one to obtain numerical results of principal eigenvalue problems for non-self-adjoint operators in such high dimensions.

This article aims to study efficient/trace optimal designs for crossover trials with multiple responses recorded from each subject in the time periods. A multivariate fixed effects model is proposed with direct and carryover effects corresponding to the multiple responses. The corresponding error dispersion matrix is chosen to be either of the proportional or the generalized Markov covariance type, permitting the existence of direct and cross-correlations within and between the multiple responses. The corresponding information matrices for direct effects under the two types of dispersions are used to determine efficient designs. The efficiency of orthogonal array designs of Type $I$ and strength $2$ is investigated for a wide choice of covariance functions, namely, Mat($0.5$), Mat($1.5$) and Mat($\infty$). To motivate these multivariate crossover designs, a gene expression dataset in a $3 \times 3$ framework is utilized.

The simulation of many complex phenomena in engineering and science requires solving expensive, high-dimensional systems of partial differential equations (PDEs). To circumvent this, reduced-order models (ROMs) have been developed to speed up computations. However, when governing equations are unknown or partially known, typically ROMs lack interpretability and reliability of the predicted solutions. In this work we present a data-driven, non-intrusive framework for building ROMs where the latent variables and dynamics are identified in an interpretable manner and uncertainty is quantified. Starting from a limited amount of high-dimensional, noisy data the proposed framework constructs an efficient ROM by leveraging variational autoencoders for dimensionality reduction along with a newly introduced, variational version of sparse identification of nonlinear dynamics (SINDy), which we refer to as Variational Identification of Nonlinear Dynamics (VINDy). In detail, the method consists of Variational Encoding of Noisy Inputs (VENI) to identify the distribution of reduced coordinates. Simultaneously, we learn the distribution of the coefficients of a pre-determined set of candidate functions by VINDy. Once trained offline, the identified model can be queried for new parameter instances and new initial conditions to compute the corresponding full-time solutions. The probabilistic setup enables uncertainty quantification as the online testing consists of Variational Inference naturally providing Certainty Intervals (VICI). In this work we showcase the effectiveness of the newly proposed VINDy method in identifying interpretable and accurate dynamical system for the R\"ossler system with different noise intensities and sources. Then the performance of the overall method - named VENI, VINDy, VICI - is tested on PDE benchmarks including structural mechanics and fluid dynamics.

Quantum machine learning requires powerful, flexible and efficiently trainable models to be successful in solving challenging problems. In this work, we present density quantum neural networks, a learning model incorporating randomisation over a set of trainable unitaries. These models generalise quantum neural networks using parameterised quantum circuits, and allow a trade-off between expressibility and efficient trainability, particularly on quantum hardware. We demonstrate the flexibility of the formalism by applying it to two recently proposed model families. The first are commuting-block quantum neural networks (QNNs) which are efficiently trainable but may be limited in expressibility. The second are orthogonal (Hamming-weight preserving) quantum neural networks which provide well-defined and interpretable transformations on data but are challenging to train at scale on quantum devices. Density commuting QNNs improve capacity with minimal gradient complexity overhead, and density orthogonal neural networks admit a quadratic-to-constant gradient query advantage with minimal to no performance loss. We conduct numerical experiments on synthetic translationally invariant data and MNIST image data with hyperparameter optimisation to support our findings. Finally, we discuss the connection to post-variational quantum neural networks, measurement-based quantum machine learning and the dropout mechanism.

Identifiability of statistical models is a key notion in unsupervised representation learning. Recent work of nonlinear independent component analysis (ICA) employs auxiliary data and has established identifiable conditions. This paper proposes a statistical model of two latent vectors with single auxiliary data generalizing nonlinear ICA, and establishes various identifiability conditions. Unlike previous work, the two latent vectors in the proposed model can have arbitrary dimensions, and this property enables us to reveal an insightful dimensionality relation among two latent vectors and auxiliary data in identifiability conditions. Furthermore, surprisingly, we prove that the indeterminacies of the proposed model has the same as \emph{linear} ICA under certain conditions: The elements in the latent vector can be recovered up to their permutation and scales. Next, we apply the identifiability theory to a statistical model for graph data. As a result, one of the identifiability conditions includes an appealing implication: Identifiability of the statistical model could depend on the maximum value of link weights in graph data. Then, we propose a practical method for identifiable graph embedding. Finally, we numerically demonstrate that the proposed method well-recovers the latent vectors and model identifiability clearly depends on the maximum value of link weights, which supports the implication of our theoretical results

We derive information-theoretic generalization bounds for supervised learning algorithms based on the information contained in predictions rather than in the output of the training algorithm. These bounds improve over the existing information-theoretic bounds, are applicable to a wider range of algorithms, and solve two key challenges: (a) they give meaningful results for deterministic algorithms and (b) they are significantly easier to estimate. We show experimentally that the proposed bounds closely follow the generalization gap in practical scenarios for deep learning.

Deep learning is usually described as an experiment-driven field under continuous criticizes of lacking theoretical foundations. This problem has been partially fixed by a large volume of literature which has so far not been well organized. This paper reviews and organizes the recent advances in deep learning theory. The literature is categorized in six groups: (1) complexity and capacity-based approaches for analyzing the generalizability of deep learning; (2) stochastic differential equations and their dynamic systems for modelling stochastic gradient descent and its variants, which characterize the optimization and generalization of deep learning, partially inspired by Bayesian inference; (3) the geometrical structures of the loss landscape that drives the trajectories of the dynamic systems; (4) the roles of over-parameterization of deep neural networks from both positive and negative perspectives; (5) theoretical foundations of several special structures in network architectures; and (6) the increasingly intensive concerns in ethics and security and their relationships with generalizability.

Machine-learning models have demonstrated great success in learning complex patterns that enable them to make predictions about unobserved data. In addition to using models for prediction, the ability to interpret what a model has learned is receiving an increasing amount of attention. However, this increased focus has led to considerable confusion about the notion of interpretability. In particular, it is unclear how the wide array of proposed interpretation methods are related, and what common concepts can be used to evaluate them. We aim to address these concerns by defining interpretability in the context of machine learning and introducing the Predictive, Descriptive, Relevant (PDR) framework for discussing interpretations. The PDR framework provides three overarching desiderata for evaluation: predictive accuracy, descriptive accuracy and relevancy, with relevancy judged relative to a human audience. Moreover, to help manage the deluge of interpretation methods, we introduce a categorization of existing techniques into model-based and post-hoc categories, with sub-groups including sparsity, modularity and simulatability. To demonstrate how practitioners can use the PDR framework to evaluate and understand interpretations, we provide numerous real-world examples. These examples highlight the often under-appreciated role played by human audiences in discussions of interpretability. Finally, based on our framework, we discuss limitations of existing methods and directions for future work. We hope that this work will provide a common vocabulary that will make it easier for both practitioners and researchers to discuss and choose from the full range of interpretation methods.