This study focuses on addressing the inverse source problem associated with the parabolic equation. We rely on sparse boundary flux data as our measurements, which are acquired from a restricted section of the boundary. While it has been established that utilizing sparse boundary flux data can enable source recovery, the presence of a limited number of observation sensors poses a challenge for accurately tracing the inverse quantity of interest. To overcome this limitation, we introduce a sampling algorithm grounded in Langevin dynamics that incorporates dynamic sensors to capture the flux information. Furthermore, we propose and discuss two distinct sensor migration strategies. Remarkably, our findings demonstrate that even with only two observation sensors at our disposal, it remains feasible to successfully reconstruct the high-dimensional unknown parameters.
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Selecting the best regularization parameter in inverse problems is a classical and yet challenging problem. Recently, data-driven approaches have become popular to tackle this challenge. These approaches are appealing since they do require less a priori knowledge, but their theoretical analysis is limited. In this paper, we propose and study a statistical machine learning approach, based on empirical risk minimization. Our main contribution is a theoretical analysis, showing that, provided with enough data, this approach can reach sharp rates while being essentially adaptive to the noise and smoothness of the problem. Numerical simulations corroborate and illustrate the theoretical findings. Our results are a step towards grounding theoretically data-driven approaches to inverse problems.
A major concern in using deep learning based generative models for document-grounded dialogs is the potential generation of responses that are not \textit{faithful} to the underlying document. Existing automated metrics used for evaluating the faithfulness of response with respect to the grounding document measure the degree of similarity between the generated response and the document's content. However, these automated metrics are far from being well aligned with human judgments. Therefore, to improve the measurement of faithfulness, we propose a new metric that utilizes (Conditional) Point-wise Mutual Information (PMI) between the generated response and the source document, conditioned on the dialogue. PMI quantifies the extent to which the document influences the generated response -- with a higher PMI indicating a more faithful response. We build upon this idea to create a new decoding technique that incorporates PMI into the response generation process to predict more faithful responses. Our experiments on the BEGIN benchmark demonstrate an improved correlation of our metric with human evaluation. We also show that our decoding technique is effective in generating more faithful responses when compared to standard decoding techniques on a set of publicly available document-grounded dialog datasets.
We introduce a framework for solving a class of parabolic partial differential equations on triangle mesh surfaces, including the Hamilton-Jacobi equation and the Fokker-Planck equation. PDE in this class often have nonlinear or stiff terms that cannot be resolved with standard methods on curved triangle meshes. To address this challenge, we leverage a splitting integrator combined with a convex optimization step to solve these PDE. Our machinery can be used to compute entropic approximation of optimal transport distances on geometric domains, overcoming the numerical limitations of the state-of-the-art method. In addition, we demonstrate the versatility of our method on a number of linear and nonlinear PDE that appear in diffusion and front propagation tasks in geometry processing.
Many important tasks of large-scale recommender systems can be naturally cast as testing multiple linear forms for noisy matrix completion. These problems, however, present unique challenges because of the subtle bias-and-variance tradeoff of and an intricate dependence among the estimated entries induced by the low-rank structure. In this paper, we develop a general approach to overcome these difficulties by introducing new statistics for individual tests with sharp asymptotics both marginally and jointly, and utilizing them to control the false discovery rate (FDR) via a data splitting and symmetric aggregation scheme. We show that valid FDR control can be achieved with guaranteed power under nearly optimal sample size requirements using the proposed methodology. Extensive numerical simulations and real data examples are also presented to further illustrate its practical merits.
We study the scheduling problem in a status update system composed of an arbitrary number of information sources with different service time distributions and weights for the purpose of minimizing the weighted sum age of information (AoI). In particular, we study open-loop schedulers which rely only on the statistics (specifically, only on the first two moments) of the source service times, in contrast to closed-loop schedulers that also make use of the actual realizations of the service times and the AoI processes in making scheduling decisions. Open-loop scheduling policies can be constructed off-line and are simpler to implement compared to their closed-loop counterparts. We consider the generate-at-will (GAW) model, and develop an analytical method to calculate the exact AoI for the probabilistic and cyclic open-loop schedulers. In both cases, the server initiates the sampling of a source and the ensuing transmission of the update packet from the source to the server in an open-loop manner; either based on a certain probability (probabilistic scheme) or according to a deterministic cyclic pattern (cyclic scheme). We derive the optimum open-loop cyclic scheduling policy in closed form for the specific case of N=2 sources and propose well-performing heuristic cyclic schedulers for general number of sources, i.e., N>2. We study the proposed cyclic schedulers against probabilistic schedulers and several existing methods in the literature to validate their effectiveness.
We present a novel formulation of structural design optimization problems specifically tailored to be solved by quantum annealing (QA). Structural design optimization aims to find the best, i.e., material-efficient yet high-performance, configuration of a structure. To this end, computational optimization strategies can be employed, where a recently evolving strategy based on quantum mechanical effects is QA. This approach requires the optimization problem to be present, e.g., as a quadratic unconstrained binary optimization (QUBO) model. Thus, we develop a novel formulation of the optimization problem. The latter typically involves an analysis model for the component. Here, we use energy minimization principles that govern the behavior of structures under applied loads. This allows us to state the optimization problem as one overall minimization problem. Next, we map this to a QUBO problem that can be immediately solved by QA. We validate the proposed approach using a size optimization problem of a compound rod under self-weight loading. To this end, we develop strategies to account for the limitations of currently available hardware and find that the presented formulation is suitable for solving structural design optimization problems through QA and, for small-scale problems, already works on today's hardware.
The recent discovery of a connection between Transformers and Modern Hopfield Networks (MHNs) has reignited the study of neural networks from a physical energy-based perspective. This paper focuses on the pivotal effect of the inverse temperature hyperparameter $\beta$ on the distribution of energy minima of the MHN. To achieve this, the distribution of energy minima is tracked in a simplified MHN in which equidistant normalised patterns are stored. This network demonstrates a phase transition at a critical temperature $\beta_{\text{c}}$, from a single global attractor towards highly pattern specific minima as $\beta$ is increased. Importantly, the dynamics are not solely governed by the hyperparameter $\beta$ but are instead determined by an effective inverse temperature $\beta_{\text{eff}}$ which also depends on the distribution and size of the stored patterns. Recognizing the role of hyperparameters in the MHN could, in the future, aid researchers in the domain of Transformers to optimise their initial choices, potentially reducing the necessity for time and energy expensive hyperparameter fine-tuning.
Many areas of machine learning and science involve large linear algebra problems, such as eigendecompositions, solving linear systems, computing matrix exponentials, and trace estimation. The matrices involved often have Kronecker, convolutional, block diagonal, sum, or product structure. In this paper, we propose a simple but general framework for large-scale linear algebra problems in machine learning, named CoLA (Compositional Linear Algebra). By combining a linear operator abstraction with compositional dispatch rules, CoLA automatically constructs memory and runtime efficient numerical algorithms. Moreover, CoLA provides memory efficient automatic differentiation, low precision computation, and GPU acceleration in both JAX and PyTorch, while also accommodating new objects, operations, and rules in downstream packages via multiple dispatch. CoLA can accelerate many algebraic operations, while making it easy to prototype matrix structures and algorithms, providing an appealing drop-in tool for virtually any computational effort that requires linear algebra. We showcase its efficacy across a broad range of applications, including partial differential equations, Gaussian processes, equivariant model construction, and unsupervised learning.
As soon as abstract mathematical computations were adapted to computation on digital computers, the problem of efficient representation, manipulation, and communication of the numerical values in those computations arose. Strongly related to the problem of numerical representation is the problem of quantization: in what manner should a set of continuous real-valued numbers be distributed over a fixed discrete set of numbers to minimize the number of bits required and also to maximize the accuracy of the attendant computations? This perennial problem of quantization is particularly relevant whenever memory and/or computational resources are severely restricted, and it has come to the forefront in recent years due to the remarkable performance of Neural Network models in computer vision, natural language processing, and related areas. Moving from floating-point representations to low-precision fixed integer values represented in four bits or less holds the potential to reduce the memory footprint and latency by a factor of 16x; and, in fact, reductions of 4x to 8x are often realized in practice in these applications. Thus, it is not surprising that quantization has emerged recently as an important and very active sub-area of research in the efficient implementation of computations associated with Neural Networks. In this article, we survey approaches to the problem of quantizing the numerical values in deep Neural Network computations, covering the advantages/disadvantages of current methods. With this survey and its organization, we hope to have presented a useful snapshot of the current research in quantization for Neural Networks and to have given an intelligent organization to ease the evaluation of future research in this area.
High spectral dimensionality and the shortage of annotations make hyperspectral image (HSI) classification a challenging problem. Recent studies suggest that convolutional neural networks can learn discriminative spatial features, which play a paramount role in HSI interpretation. However, most of these methods ignore the distinctive spectral-spatial characteristic of hyperspectral data. In addition, a large amount of unlabeled data remains an unexploited gold mine for efficient data use. Therefore, we proposed an integration of generative adversarial networks (GANs) and probabilistic graphical models for HSI classification. Specifically, we used a spectral-spatial generator and a discriminator to identify land cover categories of hyperspectral cubes. Moreover, to take advantage of a large amount of unlabeled data, we adopted a conditional random field to refine the preliminary classification results generated by GANs. Experimental results obtained using two commonly studied datasets demonstrate that the proposed framework achieved encouraging classification accuracy using a small number of data for training.
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