Optimal recovery is a mathematical framework for learning functions from observational data by adopting a worst-case perspective tied to model assumptions on the functions to be learned. Working in a finite-dimensional Hilbert space, we consider model assumptions based on approximability and observation inaccuracies modeled as additive errors bounded in $\ell_2$. We focus on the local recovery problem, which amounts to the determination of Chebyshev centers. Earlier work by Beck and Eldar presented a semidefinite recipe for the determination of Chebyshev centers. The result was valid in the complex setting only, but not necessarily in the real setting, since it relied on the S-procedure with two quadratic constraints, which offers a tight relaxation only in the complex setting. Our contribution consists in proving that this semidefinite recipe is exact in the real setting, too, at least in the particular instance where the quadratic constraints involve orthogonal projectors. Our argument exploits a previous work of ours, where exact Chebyshev centers were obtained in a different way. We conclude by stating some open questions and by commenting on other recent results in optimal recovery.
相關內容
Due to its conceptual simplicity and generality, compressive neural representation has emerged as a promising alternative to traditional compression methods for managing massive volumetric datasets. The current practice of neural compression utilizes a single large multilayer perceptron (MLP) to encode the global volume, incurring slow training and inference. This paper presents an efficient compressive neural representation (ECNR) solution for time-varying data compression, utilizing the Laplacian pyramid for adaptive signal fitting. Following a multiscale structure, we leverage multiple small MLPs at each scale for fitting local content or residual blocks. By assigning similar blocks to the same MLP via size uniformization, we enable balanced parallelization among MLPs to significantly speed up training and inference. Working in concert with the multiscale structure, we tailor a deep compression strategy to compact the resulting model. We show the effectiveness of ECNR with multiple datasets and compare it with state-of-the-art compression methods (mainly SZ3, TTHRESH, and neurcomp). The results position ECNR as a promising solution for volumetric data compression.
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.
Out-of-distribution (OOD) detection is a crucial task for ensuring the reliability and robustness of machine learning models. Recent works have shown that generative models often assign high confidence scores to OOD samples, indicating that they fail to capture the semantic information of the data. To tackle this problem, we take advantage of sample repairing and propose a novel OOD detection framework, namely SR-OOD. Our framework leverages the idea that repairing an OOD sample can reveal its semantic inconsistency with the in-distribution data. Specifically, our framework consists of two components: a sample repairing module and a detection module. The sample repairing module applies erosion to an input sample and uses a generative adversarial network to repair it. The detection module then determines whether the input sample is OOD using a distance metric. Our framework does not require any additional data or label information for detection, making it applicable to various scenarios. We conduct extensive experiments on three image datasets: CIFAR-10, CelebA, and Pokemon. The results demonstrate that our approach achieves superior performance over the state-of-the-art generative methods in OOD detection.
Sinkhorn Flow: A Continuous-Time Framework for Understanding and Generalizing the Sinkhorn Algorithm
Many problems in machine learning can be formulated as solving entropy-regularized optimal transport on the space of probability measures. The canonical approach involves the Sinkhorn iterates, renowned for their rich mathematical properties. Recently, the Sinkhorn algorithm has been recast within the mirror descent framework, thus benefiting from classical optimization theory insights. Here, we build upon this result by introducing a continuous-time analogue of the Sinkhorn algorithm. This perspective allows us to derive novel variants of Sinkhorn schemes that are robust to noise and bias. Moreover, our continuous-time dynamics not only generalize but also offer a unified perspective on several recently discovered dynamics in machine learning and mathematics, such as the "Wasserstein mirror flow" of (Deb et al. 2023) or the "mean-field Schr\"odinger equation" of (Claisse et al. 2023).
Deep Operator Network (DeepONet) is a neural network framework for learning nonlinear operators such as those from ordinary differential equations (ODEs) describing complex systems. Multiple-input deep neural operators (MIONet) extended DeepONet to allow multiple input functions in different Banach spaces. MIONet offers flexibility in training dataset grid spacing, without constraints on output location. However, it requires offline inputs and cannot handle varying sequence lengths in testing datasets, limiting its real-time application in dynamic complex systems. This work redesigns MIONet, integrating Long Short Term Memory (LSTM) to learn neural operators from time-dependent data. This approach overcomes data discretization constraints and harnesses LSTM's capability with variable-length, real-time data. Factors affecting learning performance, like algorithm extrapolation ability are presented. The framework is enhanced with uncertainty quantification through a novel Bayesian method, sampling from MIONet parameter distributions. Consequently, we develop the B-LSTM-MIONet, incorporating LSTM's temporal strengths with Bayesian robustness, resulting in a more precise and reliable model for noisy datasets.
Optimization problems involving minimization of a rank-one convex function over constraints modeling restrictions on the support of the decision variables emerge in various machine learning applications. These problems are often modeled with indicator variables for identifying the support of the continuous variables. In this paper we investigate compact extended formulations for such problems through perspective reformulation techniques. In contrast to the majority of previous work that relies on support function arguments and disjunctive programming techniques to provide convex hull results, we propose a constructive approach that exploits a hidden conic structure induced by perspective functions. To this end, we first establish a convex hull result for a general conic mixed-binary set in which each conic constraint involves a linear function of independent continuous variables and a set of binary variables. We then demonstrate that extended representations of sets associated with epigraphs of rank-one convex functions over constraints modeling indicator relations naturally admit such a conic representation. This enables us to systematically give perspective formulations for the convex hull descriptions of these sets with nonlinear separable or non-separable objective functions, sign constraints on continuous variables, and combinatorial constraints on indicator variables. We illustrate the efficacy of our results on sparse nonnegative logistic regression problems.
Existing recommender systems extract the user preference based on learning the correlation in data, such as behavioral correlation in collaborative filtering, feature-feature, or feature-behavior correlation in click-through rate prediction. However, regretfully, the real world is driven by causality rather than correlation, and correlation does not imply causation. For example, the recommender systems can recommend a battery charger to a user after buying a phone, in which the latter can serve as the cause of the former, and such a causal relation cannot be reversed. Recently, to address it, researchers in recommender systems have begun to utilize causal inference to extract causality, enhancing the recommender system. In this survey, we comprehensively review the literature on causal inference-based recommendation. At first, we present the fundamental concepts of both recommendation and causal inference as the basis of later content. We raise the typical issues that the non-causality recommendation is faced. Afterward, we comprehensively review the existing work of causal inference-based recommendation, based on a taxonomy of what kind of problem causal inference addresses. Last, we discuss the open problems in this important research area, along with interesting future works.
The existence of representative datasets is a prerequisite of many successful artificial intelligence and machine learning models. However, the subsequent application of these models often involves scenarios that are inadequately represented in the data used for training. The reasons for this are manifold and range from time and cost constraints to ethical considerations. As a consequence, the reliable use of these models, especially in safety-critical applications, is a huge challenge. Leveraging additional, already existing sources of knowledge is key to overcome the limitations of purely data-driven approaches, and eventually to increase the generalization capability of these models. Furthermore, predictions that conform with knowledge are crucial for making trustworthy and safe decisions even in underrepresented scenarios. This work provides an overview of existing techniques and methods in the literature that combine data-based models with existing knowledge. The identified approaches are structured according to the categories integration, extraction and conformity. Special attention is given to applications in the field of autonomous driving.
Since hardware resources are limited, the objective of training deep learning models is typically to maximize accuracy subject to the time and memory constraints of training and inference. We study the impact of model size in this setting, focusing on Transformer models for NLP tasks that are limited by compute: self-supervised pretraining and high-resource machine translation. We first show that even though smaller Transformer models execute faster per iteration, wider and deeper models converge in significantly fewer steps. Moreover, this acceleration in convergence typically outpaces the additional computational overhead of using larger models. Therefore, the most compute-efficient training strategy is to counterintuitively train extremely large models but stop after a small number of iterations. This leads to an apparent trade-off between the training efficiency of large Transformer models and the inference efficiency of small Transformer models. However, we show that large models are more robust to compression techniques such as quantization and pruning than small models. Consequently, one can get the best of both worlds: heavily compressed, large models achieve higher accuracy than lightly compressed, small models.
We introduce a multi-task setup of identifying and classifying entities, relations, and coreference clusters in scientific articles. We create SciERC, a dataset that includes annotations for all three tasks and develop a unified framework called Scientific Information Extractor (SciIE) for with shared span representations. The multi-task setup reduces cascading errors between tasks and leverages cross-sentence relations through coreference links. Experiments show that our multi-task model outperforms previous models in scientific information extraction without using any domain-specific features. We further show that the framework supports construction of a scientific knowledge graph, which we use to analyze information in scientific literature.
注冊地址： 北京市海淀區羊坊店路18號2幢3層301-191