In this work a general approach to compute a compressed representation of the exponential $\exp(h)$ of a high-dimensional function $h$ is presented. Such exponential functions play an important role in several problems in Uncertainty Quantification, e.g. the approximation of log-normal random fields or the evaluation of Bayesian posterior measures. Usually, these high-dimensional objects are intractable numerically and can only be accessed pointwise in sampling methods. In contrast, the proposed method constructs a functional representation of the exponential by exploiting its nature as a solution of an ordinary differential equation. The application of a Petrov--Galerkin scheme to this equation provides a tensor train representation of the solution for which we derive an efficient and reliable a posteriori error estimator. Numerical experiments with a log-normal random field and a Bayesian likelihood illustrate the performance of the approach in comparison to other recent low-rank representations for the respective applications. Although the present work considers only a specific differential equation, the presented method can be applied in a more general setting. We show that the composition of a generic holonomic function and a high-dimensional function corresponds to a differential equation that can be used in our method. Moreover, the differential equation can be modified to adapt the norm in the a posteriori error estimates to the problem at hand.
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Bayesian model selection provides a powerful framework for objectively comparing models directly from observed data, without reference to ground truth data. However, Bayesian model selection requires the computation of the marginal likelihood (model evidence), which is computationally challenging, prohibiting its use in many high-dimensional Bayesian inverse problems. With Bayesian imaging applications in mind, in this work we present the proximal nested sampling methodology to objectively compare alternative Bayesian imaging models for applications that use images to inform decisions under uncertainty. The methodology is based on nested sampling, a Monte Carlo approach specialised for model comparison, and exploits proximal Markov chain Monte Carlo techniques to scale efficiently to large problems and to tackle models that are log-concave and not necessarily smooth (e.g., involving l_1 or total-variation priors). The proposed approach can be applied computationally to problems of dimension O(10^6) and beyond, making it suitable for high-dimensional inverse imaging problems. It is validated on large Gaussian models, for which the likelihood is available analytically, and subsequently illustrated on a range of imaging problems where it is used to analyse different choices of dictionary and measurement model.
The learning speed of feed-forward neural networks is notoriously slow and has presented a bottleneck in deep learning applications for several decades. For instance, gradient-based learning algorithms, which are used extensively to train neural networks, tend to work slowly when all of the network parameters must be iteratively tuned. To counter this, both researchers and practitioners have tried introducing randomness to reduce the learning requirement. Based on the original construction of Igelnik and Pao, single layer neural-networks with random input-to-hidden layer weights and biases have seen success in practice, but the necessary theoretical justification is lacking. In this paper, we begin to fill this theoretical gap. We provide a (corrected) rigorous proof that the Igelnik and Pao construction is a universal approximator for continuous functions on compact domains, with approximation error decaying asymptotically like $O(1/\sqrt{n})$ for the number $n$ of network nodes. We then extend this result to the non-asymptotic setting, proving that one can achieve any desired approximation error with high probability provided $n$ is sufficiently large. We further adapt this randomized neural network architecture to approximate functions on smooth, compact submanifolds of Euclidean space, providing theoretical guarantees in both the asymptotic and non-asymptotic forms. Finally, we illustrate our results on manifolds with numerical experiments.
Edge training of Deep Neural Networks (DNNs) is a desirable goal for continuous learning; however, it is hindered by the enormous computational power required by training. Hardware approximate multipliers have shown their effectiveness for gaining resource-efficiency in DNN inference accelerators; however, training with approximate multipliers is largely unexplored. To build resource efficient accelerators with approximate multipliers supporting DNN training, a thorough evaluation of training convergence and accuracy for different DNN architectures and different approximate multipliers is needed. This paper presents ApproxTrain, an open-source framework that allows fast evaluation of DNN training and inference using simulated approximate multipliers. ApproxTrain is as user-friendly as TensorFlow (TF) and requires only a high-level description of a DNN architecture along with C/C++ functional models of the approximate multiplier. We improve the speed of the simulation at the multiplier level by using a novel LUT-based approximate floating-point (FP) multiplier simulator on GPU (AMSim). ApproxTrain leverages CUDA and efficiently integrates AMSim into the TensorFlow library, in order to overcome the absence of native hardware approximate multiplier in commercial GPUs. We use ApproxTrain to evaluate the convergence and accuracy of DNN training with approximate multipliers for small and large datasets (including ImageNet) using LeNets and ResNets architectures. The evaluations demonstrate similar convergence behavior and negligible change in test accuracy compared to FP32 and bfloat16 multipliers. Compared to CPU-based approximate multiplier simulations in training and inference, the GPU-accelerated ApproxTrain is more than 2500x faster. Based on highly optimized closed-source cuDNN/cuBLAS libraries with native hardware multipliers, the original TensorFlow is only 8x faster than ApproxTrain.
Active inference provides a general framework for behavior and learning in autonomous agents. It states that an agent will attempt to minimize its variational free energy, defined in terms of beliefs over observations, internal states and policies. Traditionally, every aspect of a discrete active inference model must be specified by hand, i.e. by manually defining the hidden state space structure, as well as the required distributions such as likelihood and transition probabilities. Recently, efforts have been made to learn state space representations automatically from observations using deep neural networks. In this paper, we present a novel approach of learning state spaces using quantum physics-inspired tensor networks. The ability of tensor networks to represent the probabilistic nature of quantum states as well as to reduce large state spaces makes tensor networks a natural candidate for active inference. We show how tensor networks can be used as a generative model for sequential data. Furthermore, we show how one can obtain beliefs from such a generative model and how an active inference agent can use these to compute the expected free energy. Finally, we demonstrate our method on the classic T-maze environment.
Although there is an extensive literature on the eigenvalues of high-dimensional sample covariance matrices, much of it is specialized to Mar\v{c}enko-Pastur (MP) models -- in which observations are represented as linear transformations of random vectors with independent entries. By contrast, less is known in the context of elliptical models, which violate the independence structure of MP models and exhibit quite different statistical phenomena. In particular, very little is known about the scope of bootstrap methods for doing inference with spectral statistics in high-dimensional elliptical models. To fill this gap, we show how a bootstrap approach developed previously for MP models can be extended to handle the different properties of elliptical models. Within this setting, our main theoretical result guarantees that the proposed method consistently approximates the distributions of linear spectral statistics, which play a fundamental role in multivariate analysis. Lastly, we provide empirical results showing that the proposed method also performs well for a variety of nonlinear spectral statistics.
Very often, in the course of uncertainty quantification tasks or data analysis, one has to deal with high-dimensional random variables (RVs). A high-dimensional RV can be described by its probability density (pdf) and/or by the corresponding probability characteristic functions (pcf), or by a polynomial chaos (PCE) or similar expansion. Here the interest is mainly to compute characterisations like the entropy, or relations between two distributions, like their Kullback-Leibler divergence. These are all computed from the pdf, which is often not available directly, and it is a computational challenge to even represent it in a numerically feasible fashion in case the dimension $d$ is even moderately large. In this regard, we propose to represent the density by a high order tensor product, and approximate this in a low-rank format. We show how to go from the pcf or functional representation to the pdf. This allows us to reduce the computational complexity and storage cost from an exponential to a linear. The characterisations such as entropy or the $f$-divergences need the possibility to compute point-wise functions of the pdf. This normally rather trivial task becomes more difficult when the pdf is approximated in a low-rank tensor format, as the point values are not directly accessible any more. The data is considered as an element of a high order tensor space. The considered algorithms are independent of the representation of the data as a tensor. All that we require is that the data can be considered as an element of an associative, commutative algebra with an inner product. Such an algebra is isomorphic to a commutative sub-algebra of the usual matrix algebra, allowing the use of matrix algorithms to accomplish the mentioned tasks.
In this paper, we present a predictor-corrector strategy for constructing rank-adaptive dynamical low-rank approximations (DLRAs) of matrix-valued ODE systems. The strategy is a compromise between (i) low-rank step-truncation approaches that alternately evolve and compress solutions and (ii) strict DLRA approaches that augment the low-rank manifold using subspaces generated locally in time by the DLRA integrator. The strategy is based on an analysis of the error between a forward temporal update into the ambient full-rank space, which is typically computed in a step-truncation approach before re-compressing, and the standard DLRA update, which is forced to live in a low-rank manifold. We use this error, without requiring its full-rank representation, to correct the DLRA solution. A key ingredient for maintaining a low-rank representation of the error is a randomized singular value decomposition (SVD), which introduces some degree of stochastic variability into the implementation. The strategy is formulated and implemented in the context of discontinuous Galerkin spatial discretizations of partial differential equations and applied to several versions of DLRA methods found in the literature, as well as a new variant. Numerical experiments comparing the predictor-corrector strategy to other methods demonstrate robustness to overcome short-comings of step truncation or strict DLRA approaches: the former may require more memory than is strictly needed while the latter may miss transients solution features that cannot be recovered. The effect of randomization, tolerances, and other implementation parameters is also explored.
Chirp signal models and their generalizations have been used to model many natural and man-made phenomena in signal processing and time series literature. In recent times, several methods have been proposed for parameter estimation of these models. These methods however are either statistically sub-optimal or computationally burdensome, specially for two dimensional (2D) chirp models. In this paper, we consider the problem of parameter estimation of 2D chirp models and propose a computationally efficient estimator and establish asymptotic theoretical properties of the proposed estimators. And the proposed estimators are observed to have the same rates of convergence as the least squares estimators (LSEs). Furthermore, the proposed estimators of chirp rate parameters are shown to be asymptotically optimal. Extensive and detailed numerical simulations are conducted, which support theoretical results of the proposed estimators.
We consider the identification of spatially distributed parameters under $H^1$ regularization. Solving the associated minimization problem by Gauss-Newton iteration results in linearized problems to be solved in each step that can be cast as boundary value problems involving a low-rank modification of the Laplacian. Using algebraic multigrid as a fast Laplace solver, the Sherman-Morrison-Woodbury formula can be employed to construct a preconditioner for these linear problems which exhibits excellent scaling w.r.t. the relevant problem parameters. We first develop this approach in the functional setting, thus obtaining a consistent methodology for selecting boundary conditions that arise from the $H^1$ regularization. We then construct a method for solving the discrete linear systems based on combining any fast Poisson solver with the Woodbury formula. The efficacy of this method is then demonstrated with scaling experiments. These are carried out for a common nonlinear parameter identification problem arising in electrical resistivity tomography.
Sampling methods (e.g., node-wise, layer-wise, or subgraph) has become an indispensable strategy to speed up training large-scale Graph Neural Networks (GNNs). However, existing sampling methods are mostly based on the graph structural information and ignore the dynamicity of optimization, which leads to high variance in estimating the stochastic gradients. The high variance issue can be very pronounced in extremely large graphs, where it results in slow convergence and poor generalization. In this paper, we theoretically analyze the variance of sampling methods and show that, due to the composite structure of empirical risk, the variance of any sampling method can be decomposed into \textit{embedding approximation variance} in the forward stage and \textit{stochastic gradient variance} in the backward stage that necessities mitigating both types of variance to obtain faster convergence rate. We propose a decoupled variance reduction strategy that employs (approximate) gradient information to adaptively sample nodes with minimal variance, and explicitly reduces the variance introduced by embedding approximation. We show theoretically and empirically that the proposed method, even with smaller mini-batch sizes, enjoys a faster convergence rate and entails a better generalization compared to the existing methods.
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