Implementation of many statistical methods for large, multivariate data sets requires one to solve a linear system that, depending on the method, is of the dimension of the number of observations or each individual data vector. This is often the limiting factor in scaling the method with data size and complexity. In this paper we illustrate the use of Krylov subspace methods to address this issue in a statistical solution to a source separation problem in cosmology where the data size is prohibitively large for direct solution of the required system. Two distinct approaches, adapted from techniques in the literature, are described: one that uses the method of conjugate gradients directly to the Kronecker-structured problem and another that reformulates the system as a Sylvester matrix equation. We show that both approaches produce an accurate solution within an acceptable computation time and with practical memory requirements for the data size that is currently available.
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We evaluate the robustness of several large language models on multiple datasets. Robustness here refers to the relative insensitivity of the model's answers to meaning-preserving variants of their input. Benchmark datasets are constructed by introducing naturally-occurring, non-malicious perturbations, or by generating semantically equivalent paraphrases of input questions or statements. We further propose a novel metric for assessing a model robustness, and demonstrate its benefits in the non-adversarial scenario by empirical evaluation of several models on the created datasets.
Our focus is on simulating the dynamics of non-interacting particles, which, under certain assumptions, can be formally described by the Dean-Kawasaki equation. The Dean-Kawasaki equation can be solved numerically using standard finite volume methods. However, the numerical approximation implicitly requires a sufficiently large number of particles to ensure the positivity of the solution and accurate approximation of the stochastic flux. To address this challenge, we extend hybrid algorithms for particle systems to scenarios where the density is low. The aim is to create a hybrid algorithm that switches from a finite volume discretization to a particle-based method when the particle density falls below a certain threshold. We develop criteria for determining this threshold by comparing higher-order statistics obtained from the finite volume method with particle simulations. We then demonstrate the use of the resulting criteria for dynamic adaptation in both two- and three-dimensional spatial settings.
Many models require integrals of high-dimensional functions: for instance, to obtain marginal likelihoods. Such integrals may be intractable, or too expensive to compute numerically. Instead, we can use the Laplace approximation (LA). The LA is exact if the function is proportional to a normal density; its effectiveness therefore depends on the function's true shape. Here, we propose the use of the probabilistic numerical framework to develop a diagnostic for the LA and its underlying shape assumptions, modelling the function and its integral as a Gaussian process and devising a "test" by conditioning on a finite number of function values. The test is decidedly non-asymptotic and is not intended as a full substitute for numerical integration - rather, it is simply intended to test the feasibility of the assumptions underpinning the LA with as minimal computation. We discuss approaches to optimize and design the test, apply it to known sample functions, and highlight the challenges of high dimensions.
This paper addresses nonparametric estimation of nonlinear multivariate Hawkes processes, where the interaction functions are assumed to lie in a reproducing kernel Hilbert space (RKHS). Motivated by applications in neuroscience, the model allows complex interaction functions, in order to express exciting and inhibiting effects, but also a combination of both (which is particularly interesting to model the refractory period of neurons), and considers in return that conditional intensities are rectified by the ReLU function. The latter feature incurs several methodological challenges, for which workarounds are proposed in this paper. In particular, it is shown that a representer theorem can be obtained for approximated versions of the log-likelihood and the least-squares criteria. Based on it, we propose an estimation method, that relies on two simple approximations (of the ReLU function and of the integral operator). We provide an approximation bound, justifying the negligible statistical effect of these approximations. Numerical results on synthetic data confirm this fact as well as the good asymptotic behavior of the proposed estimator. It also shows that our method achieves a better performance compared to related nonparametric estimation techniques and suits neuronal applications.
We investigate popular resampling methods for estimating the uncertainty of statistical models, such as subsampling, bootstrap and the jackknife, and their performance in high-dimensional supervised regression tasks. We provide a tight asymptotic description of the biases and variances estimated by these methods in the context of generalized linear models, such as ridge and logistic regression, taking the limit where the number of samples $n$ and dimension $d$ of the covariates grow at a comparable fixed rate $\alpha\!=\! n/d$. Our findings are three-fold: i) resampling methods are fraught with problems in high dimensions and exhibit the double-descent-like behavior typical of these situations; ii) only when $\alpha$ is large enough do they provide consistent and reliable error estimations (we give convergence rates); iii) in the over-parametrized regime $\alpha\!<\!1$ relevant to modern machine learning practice, their predictions are not consistent, even with optimal regularization.
Neural-networks-driven intelligent data-plane (NN-driven IDP) is becoming an emerging topic for excellent accuracy and high performance. Meanwhile we argue that NN-driven IDP should satisfy three design goals: the flexibility to support various NNs models, the low-latency-high-throughput inference performance, and the data-plane-unawareness harming no performance and functionality. Unfortunately, existing work either over-modify NNs for IDP, or insert inline pipelined accelerators into the data-plane, failing to meet the flexibility and unawareness goals. In this paper, we propose Kaleidoscope, a flexible and high-performance co-processor located at the bypass of the data-plane. To address the challenge of meeting three design goals, three key techniques are presented. The programmable run-to-completion accelerators are developed for flexible inference. To further improve performance, we design a scalable inference engine which completes low-latency and low-cost inference for the mouse flows, and perform complex NNs with high-accuracy for the elephant flows. Finally, raw-bytes-based NNs are introduced, which help to achieve unawareness. We prototype Kaleidoscope on both FPGA and ASIC library. In evaluation on six NNs models, Kaleidoscope reaches 256-352 ns inference latency and 100 Gbps throughput with negligible influence on the data-plane. The on-board tested NNs perform state-of-the-art accuracy among other NN-driven IDP, exhibiting the the significant impact of flexibility on enhancing traffic analysis accuracy.
The emergence of intelligent applications has fostered the development of a task-oriented communication paradigm, where a comprehensive, universal, and practical metric is crucial for unleashing the potential of this paradigm. To this end, we introduce an innovative metric, the Task-oriented Age of Information (TAoI), to measure whether the content of information is relevant to the system task, thereby assisting the system in efficiently completing designated tasks. Also, we study the TAoI in a remote monitoring system, whose task is to identify target images and transmit them for subsequent analysis. We formulate the dynamic transmission problem as a Semi-Markov Decision Process (SMDP) and transform it into an equivalent Markov Decision Process (MDP) to minimize TAoI and find the optimal transmission policy. Furthermore, we demonstrate that the optimal strategy is a threshold-based policy regarding TAoI and propose a relative value iteration algorithm based on the threshold structure to obtain the optimal transmission policy. Finally, simulation results show the superior performance of the optimal transmission policy compared to the baseline policies.
This paper addresses the problem of scheduling non-preemptive tasks with release jitter and execution time variation on a uniprocessor. We show that the schedulability analysis based on schedule graph generation, proposed by Nasri and Brandenburg [RTSS 2017], produces negative results when it could be easily avoided by slightly reformalizing the notion of non-work-conserving policies. In this work, we develop a schedulability analysis that constructs the schedule graph using new job-eligibility rules and is exact and sustainable for both work-conserving and enhanced formalization of non-work-conserving policies. Besides, the experimental evaluation shows that our schedulability analysis is substantially faster.
Recently, graph neural networks have been gaining a lot of attention to simulate dynamical systems due to their inductive nature leading to zero-shot generalizability. Similarly, physics-informed inductive biases in deep-learning frameworks have been shown to give superior performance in learning the dynamics of physical systems. There is a growing volume of literature that attempts to combine these two approaches. Here, we evaluate the performance of thirteen different graph neural networks, namely, Hamiltonian and Lagrangian graph neural networks, graph neural ODE, and their variants with explicit constraints and different architectures. We briefly explain the theoretical formulation highlighting the similarities and differences in the inductive biases and graph architecture of these systems. We evaluate these models on spring, pendulum, gravitational, and 3D deformable solid systems to compare the performance in terms of rollout error, conserved quantities such as energy and momentum, and generalizability to unseen system sizes. Our study demonstrates that GNNs with additional inductive biases, such as explicit constraints and decoupling of kinetic and potential energies, exhibit significantly enhanced performance. Further, all the physics-informed GNNs exhibit zero-shot generalizability to system sizes an order of magnitude larger than the training system, thus providing a promising route to simulate large-scale realistic systems.
Although Transformer-based methods have significantly improved state-of-the-art results for long-term series forecasting, they are not only computationally expensive but more importantly, are unable to capture the global view of time series (e.g. overall trend). To address these problems, we propose to combine Transformer with the seasonal-trend decomposition method, in which the decomposition method captures the global profile of time series while Transformers capture more detailed structures. To further enhance the performance of Transformer for long-term prediction, we exploit the fact that most time series tend to have a sparse representation in well-known basis such as Fourier transform, and develop a frequency enhanced Transformer. Besides being more effective, the proposed method, termed as Frequency Enhanced Decomposed Transformer ({\bf FEDformer}), is more efficient than standard Transformer with a linear complexity to the sequence length. Our empirical studies with six benchmark datasets show that compared with state-of-the-art methods, FEDformer can reduce prediction error by $14.8\%$ and $22.6\%$ for multivariate and univariate time series, respectively. the code will be released soon.
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