The Universal Approximation Theorem posits that neural networks can theoretically possess unlimited approximation capacity with a suitable activation function and a freely chosen or trained set of parameters. However, a more practical scenario arises when these neural parameters, especially the nonlinear weights and biases, are bounded. This leads us to question: \textbf{Does the approximation capacity of a neural network remain universal, or does it have a limit when the parameters are practically bounded? And if it has a limit, how can it be measured?} Our theoretical study indicates that while universal approximation is theoretically feasible, in practical numerical scenarios, Deep Neural Networks (DNNs) with any analytic activation functions (such as Tanh and Sigmoid) can only be approximated by a finite-dimensional vector space under a bounded nonlinear parameter space (NP space), whether in a continuous or discrete sense. Based on this study, we introduce the concepts of \textit{$\epsilon$ outer measure} and \textit{Numerical Span Dimension (NSdim)} to quantify the approximation capacity limit of a family of networks both theoretically and practically. Furthermore, drawing on our new theoretical study and adopting a fresh perspective, we strive to understand the relationship between back-propagation neural networks and random parameter networks (such as the Extreme Learning Machine (ELM)) with both finite and infinite width. We also aim to provide fresh insights into regularization, the trade-off between width and depth, parameter space, width redundancy, condensation, and other related important issues.

This study extends the examination of the Efficient-Market Hypothesis in Bitcoin market during a five year fluctuation period, from September 1 2017 to September 1 2022, by analyzing 28,739,514 qualified tweets containing the targeted topic "Bitcoin". Unlike previous studies, we extracted fundamental keywords as an informative proxy for carrying out the study of the EMH in the Bitcoin market rather than focusing on sentiment analysis, information volume, or price data. We tested market efficiency in hourly, 4-hourly, and daily time periods to understand the speed and accuracy of market reactions towards the information within different thresholds. A sequence of machine learning methods and textual analyses were used, including measurements of distances of semantic vector spaces of information, keywords extraction and encoding model, and Light Gradient Boosting Machine (LGBM) classifiers. Our results suggest that 78.06% (83.08%), 84.63% (87.77%), and 94.03% (94.60%) of hourly, 4-hourly, and daily bullish (bearish) market movements can be attributed to public information within organic tweets.

We study feedforward neural networks with inputs from a topological vector space (TVS-FNNs). Unlike traditional feedforward neural networks, TVS-FNNs can process a broader range of inputs, including sequences, matrices, functions and more. We prove a universal approximation theorem for TVS-FNNs, which demonstrates their capacity to approximate any continuous function defined on this expanded input space.

In the field of eXplainable AI (XAI) in language models, the progression from local explanations of individual decisions to global explanations with high-level concepts has laid the groundwork for mechanistic interpretability, which aims to decode the exact operations. However, this paradigm has not been adequately explored in image models, where existing methods have primarily focused on classspecific interpretations. This paper introduces a novel approach to systematically trace the entire pathway from input through all intermediate layers to the final output within the whole dataset. We utilize Pointwise Feature Vectors (PFVs) and Effective Receptive Fields (ERFs) to decompose model embeddings into interpretable Concept Vectors. Then, we calculate the relevance between concept vectors with our Generalized Integrated Gradients (GIG), enabling a comprehensive, dataset-wide analysis of model behavior. In the field of eXplainable AI (XAI), efforts have historically transitioned from Local explanation to Global explanation to Mechanistic Interpretability. While local explanation methods including Selvaraju et al. (2016); Montavon et al. (2017); Sundararajan et al. (2017); Han et al. (2024) have focused on explaining specific decisions for individual instances, global explanation methods seek to uncover overall patterns and behaviors applicable across the entire dataset (Wu et al., 2022; Xuanyuan et al., 2023; Singh et al., 2024).

Recent advancements in function calling and tool use have significantly enhanced the capabilities of large language models (LLMs) by enabling them to interact with external information sources and execute complex tasks. However, the limited context window of LLMs presents challenges when a large number of tools are available, necessitating efficient methods to manage prompt length and maintain accuracy. Existing approaches, such as fine-tuning LLMs or leveraging their reasoning capabilities, either require frequent retraining or incur significant latency overhead. A more efficient solution involves training smaller models to retrieve the most relevant tools for a given query, although this requires high quality, domain-specific data. To address those challenges, we present a novel framework for generating synthetic data for tool retrieval applications and an efficient data-driven tool retrieval strategy using small encoder models. Empowered by LLMs, we create ToolBank, a new tool retrieval dataset that reflects real human user usages. For tool retrieval methodologies, we propose novel approaches: (1) Tool2Vec: usage-driven tool embedding generation for tool retrieval, (2) ToolRefiner: a staged retrieval method that iteratively improves the quality of retrieved tools, and (3) MLC: framing tool retrieval as a multi-label classification problem. With these new methods, we achieve improvements of up to 27.28 in Recall@K on the ToolBench dataset and 30.5 in Recall@K on ToolBank. Additionally, we present further experimental results to rigorously validate our methods. Our code is available at \url{https://github.com/SqueezeAILab/Tool2Vec}

Approximate Nearest Neighbor Search (ANNS) is a fundamental and critical component in many applications, including recommendation systems and large language model-based applications. With the advancement of multimodal neural models, which transform data from different modalities into a shared high-dimensional space as feature vectors, cross-modal ANNS aims to use the data vector from one modality (e.g., texts) as the query to retrieve the most similar items from another (e.g., images or videos). However, there is an inherent distribution gap between embeddings from different modalities, and cross-modal queries become Out-of-Distribution (OOD) to the base data. Consequently, state-of-the-art ANNS approaches suffer poor performance for OOD workloads. In this paper, we quantitatively analyze the properties of the OOD workloads to gain an understanding of their ANNS efficiency. Unlike single-modal workloads, we reveal OOD queries spatially deviate from base data, and the k-nearest neighbors of an OOD query are distant from each other in the embedding space. The property breaks the assumptions of existing ANNS approaches and mismatches their design for efficient search. With insights from the OOD workloads, we propose pRojected bipartite Graph (RoarGraph), an efficient ANNS graph index built under the guidance of query distribution. Extensive experiments show that RoarGraph significantly outperforms state-of-the-art approaches on modern cross-modal datasets, achieving up to 3.56x faster search speed at a 90% recall rate for OOD queries.

The Gromov-Wasserstein (GW) distance is a powerful tool for comparing metric measure spaces which has found broad applications in data science and machine learning. Driven by the need to analyze datasets whose objects have increasingly complex structure (such as node and edge-attributed graphs), several variants of GW distance have been introduced in the recent literature. With a view toward establishing a general framework for the theory of GW-like distances, this paper considers a vast generalization of the notion of a metric measure space: for an arbitrary metric space $Z$, we define a $Z$-network to be a measure space endowed with a kernel valued in $Z$. We introduce a method for comparing $Z$-networks by defining a generalization of GW distance, which we refer to as $Z$-Gromov-Wasserstein ($Z$-GW) distance. This construction subsumes many previously known metrics and offers a unified approach to understanding their shared properties. The paper demonstrates that the $Z$-GW distance defines a metric on the space of $Z$-networks which retains desirable properties of $Z$, such as separability, completeness, and geodesicity. Many of these properties were unknown for existing variants of GW distance that fall under our framework. Our focus is on foundational theory, but our results also include computable lower bounds and approximations of the distance which will be useful for practical applications.

We study carbon-aware spatiotemporal workload management, which seeks to address the growing environmental impact of data centers. We formalize this as an online problem called spatiotemporal online allocation with deadline constraints ($\mathsf{SOAD}$), in which an online player completes a workload (e.g., a batch compute job) by moving and scheduling the workload across a network subject to a deadline $T$. At each time step, a service cost function is revealed, representing, e.g., the carbon intensity of servicing a workload at each location, and the player must irrevocably decide the current allocation. Furthermore, whenever the player moves the allocation, it incurs a movement cost defined by a metric space $(X,d)$ that captures, e.g., the overhead of migrating a compute job. $\mathsf{SOAD}$ formalizes the open problem of combining general metrics and deadline constraints in the online algorithms literature, unifying problems such as metrical task systems and online search. We propose a competitive algorithm for $\mathsf{SOAD}$ along with a matching lower bound that proves it is optimal. Our main algorithm, ${\rm C{\scriptsize ARBON}C{\scriptsize LIPPER}}$, is a learning-augmented algorithm that takes advantage of predictions (e.g., carbon intensity forecasts) and achieves an optimal consistency-robustness trade-off. We evaluate our proposed algorithms for carbon-aware spatiotemporal workload management on a simulated global data center network, showing that ${\rm C{\scriptsize ARBON}C{\scriptsize LIPPER}}$ significantly improves performance compared to baseline methods and delivers meaningful carbon reductions.

The increasing prevalence of graph-structured data across various domains has intensified greater interest in graph classification tasks. While numerous sophisticated graph learning methods have emerged, their complexity often hinders practical implementation. In this article, we address this challenge by proposing a method that constructs feature vectors based on fundamental graph structural properties. We demonstrate that these features, despite their simplicity, are powerful enough to capture the intrinsic characteristics of graphs within the same class. We explore the efficacy of our approach using three distinct machine learning methods, highlighting how our feature-based classification leverages the inherent structural similarities of graphs within the same class to achieve accurate classification. A key advantage of our approach is its simplicity, which makes it accessible and adaptable to a broad range of applications, including social network analysis, bioinformatics, and cybersecurity. Furthermore, we conduct extensive experiments to validate the performance of our method, showing that it not only reveals a competitive performance but in some cases surpasses the accuracy of more complex, state-of-the-art techniques. Our findings suggest that a focus on fundamental graph features can provide a robust and efficient alternative for graph classification, offering significant potential for both research and practical applications.

Advancements in modern science have led to the increasing availability of non-Euclidean data in metric spaces. This paper addresses the challenge of modeling relationships between non-Euclidean responses and multivariate Euclidean predictors. We propose a flexible regression model capable of handling high-dimensional predictors without imposing parametric assumptions. Two primary challenges are addressed: the curse of dimensionality in nonparametric regression and the absence of linear structure in general metric spaces. The former is tackled using deep neural networks, while for the latter we demonstrate the feasibility of mapping the metric space where responses reside to a low-dimensional Euclidean space using manifold learning. We introduce a reverse mapping approach, employing local Fr\'echet regression, to map the low-dimensional manifold representations back to objects in the original metric space. We develop a theoretical framework, investigating the convergence rate of deep neural networks under dependent sub-Gaussian noise with bias. The convergence rate of the proposed regression model is then obtained by expanding the scope of local Fr\'echet regression to accommodate multivariate predictors in the presence of errors in predictors. Simulations and case studies show that the proposed model outperforms existing methods for non-Euclidean responses, focusing on the special cases of probability measures and networks.