Topological data analysis (TDA) approaches are becoming increasingly popular for studying the dependence patterns in multivariate time series data. In particular, various dependence patterns in brain networks may be linked to specific tasks and cognitive processes, which can be altered by various neurological impairments such as epileptic seizures. Existing TDA approaches rely on the notion of distance between data points that is symmetric by definition for building graph filtrations. For brain dependence networks, this is a major limitation that constrains practitioners to using only symmetric dependence measures, such as correlations or coherence. However, it is known that the brain dependence network may be very complex and can contain a directed flow of information from one brain region to another. Such dependence networks are usually captured by more advanced measures of dependence such as partial directed coherence, which is a Granger causality based dependence measure. These dependence measures will result in a non-symmetric distance function, especially during epileptic seizures. In this paper we propose to solve this limitation by decomposing the weighted connectivity network into its symmetric and anti-symmetric components using matrix decomposition and comparing the anti-symmetric component prior to and post seizure. Our analysis of epileptic seizure EEG data shows promising results.
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In many applications, piecewise continuous functions are commonly interpolated over meshes. However, accurate high-order manipulations of such functions can be challenging due to potential spurious oscillations known as the Gibbs phenomena. To address this challenge, we propose a novel approach, Robust Discontinuity Indicators (RDI), which can efficiently and reliably detect both C^{0} and C^{1} discontinuities for node-based and cell-averaged values. We present a detailed analysis focusing on its derivation and the dual-thresholding strategy. A key advantage of RDI is its ability to handle potential inaccuracies associated with detecting discontinuities on non-uniform meshes, thanks to its innovative discontinuity indicators. We also extend the applicability of RDI to handle general surfaces with boundaries, features, and ridge points, thereby enhancing its versatility and usefulness in various scenarios. To demonstrate the robustness of RDI, we conduct a series of experiments on non-uniform meshes and general surfaces, and compare its performance with some alternative methods. By addressing the challenges posed by the Gibbs phenomena and providing reliable detection of discontinuities, RDI opens up possibilities for improved approximation and analysis of piecewise continuous functions, such as in data remap.
We consider statistical inference of equality-constrained stochastic nonlinear optimization problems. We develop a fully online stochastic sequential quadratic programming (StoSQP) method to solve the problems, which can be regarded as applying Newton's method to the first-order optimality conditions (i.e., the KKT conditions). Motivated by recent designs of numerical second-order methods, we allow StoSQP to adaptively select any random stepsize $\bar{\alpha}_t$, as long as $\beta_t\leq \bar{\alpha}_t \leq \beta_t+\chi_t$, for some control sequences $\beta_t$ and $\chi_t=o(\beta_t)$. To reduce the dominant computational cost of second-order methods, we additionally allow StoSQP to inexactly solve quadratic programs via efficient randomized iterative solvers that utilize sketching techniques. Notably, we do not require the approximation error to diminish as iteration proceeds. For the developed method, we show that under mild assumptions (i) computationally, it can take at most $O(1/\epsilon^4)$ iterations (same as samples) to attain $\epsilon$-stationarity; (ii) statistically, its primal-dual sequence $1/\sqrt{\beta_t}\cdot (x_t - x^\star, \lambda_t - \lambda^\star)$ converges to a mean-zero Gaussian distribution with a nontrivial covariance matrix depending on the underlying sketching distribution. Additionally, we establish the almost-sure convergence rate of the iterate $(x_t, \lambda_t)$ along with the Berry-Esseen bound; the latter quantitatively measures the convergence rate of the distribution function. We analyze a plug-in limiting covariance matrix estimator, and demonstrate the performance of the method both on benchmark nonlinear problems in CUTEst test set and on linearly/nonlinearly constrained regression problems.
Linear mixed models (LMMs) are suitable for clustered data and are common in biometrics, medicine, survey statistics and many other fields. In those applications it is essential to carry out a valid inference after selecting a subset of the available variables. We construct confidence sets for the fixed effects in Gaussian LMMs that are based on Lasso-type estimators. Aside from providing confidence regions, this also allows to quantify the joint uncertainty of both variable selection and parameter estimation in the procedure. To show that the resulting confidence sets for the fixed effects are uniformly valid over the parameter spaces of both the regression coefficients and the covariance parameters, we also prove the novel result on uniform Cramer consistency of the restricted maximum likelihood (REML) estimators of the covariance parameters. The superiority of the constructed confidence sets to naive post-selection procedures is validated in simulations and illustrated with a study of the acid neutralization capacity of lakes in the United States.
Classical recommender systems often assume that historical data are stationary and fail to account for the dynamic nature of user preferences, limiting their ability to provide reliable recommendations in time-sensitive settings. This assumption is particularly problematic in finance, where financial products exhibit continuous changes in valuations, leading to frequent shifts in client interests. These evolving interests, summarized in the past client-product interactions, see their utility fade over time with a degree that might differ from one client to another. To address this challenge, we propose a time-dependent collaborative filtering algorithm that can adaptively discount distant client-product interactions using personalized decay functions. Our approach is designed to handle the non-stationarity of financial data and produce reliable recommendations by modeling the dynamic collaborative signals between clients and products. We evaluate our method using a proprietary dataset from BNP Paribas and demonstrate significant improvements over state-of-the-art benchmarks from relevant literature. Our findings emphasize the importance of incorporating time explicitly in the model to enhance the accuracy of financial product recommendation.
We present SEIF, a methodology that combines static analysis with symbolic execution to verify and explicate information flow paths in a hardware design. SEIF begins with a statically built model of the information flow through a design and uses guided symbolic execution to recognize and eliminate non-flows with high precision or to find corresponding paths through the design state for true flows. We evaluate SEIF on two open-source CPUs, an AES core, and the AKER access control module. SEIF can exhaustively explore 10-12 clock cycles deep in 4-6 seconds on average, and can automatically account for 86-90% of the paths in the statically built model. Additionally, SEIF can be used to find multiple violating paths for security properties, providing a new angle for security verification.
Large language models (LLMs) have significantly advanced the field of natural language processing (NLP), providing a highly useful, task-agnostic foundation for a wide range of applications. The great promise of LLMs as general task solvers motivated people to extend their functionality largely beyond just a ``chatbot'', and use it as an assistant or even replacement for domain experts and tools in specific domains such as healthcare, finance, and education. However, directly applying LLMs to solve sophisticated problems in specific domains meets many hurdles, caused by the heterogeneity of domain data, the sophistication of domain knowledge, the uniqueness of domain objectives, and the diversity of the constraints (e.g., various social norms, cultural conformity, religious beliefs, and ethical standards in the domain applications). To fill such a gap, explosively-increase research, and practices have been conducted in very recent years on the domain specialization of LLMs, which, however, calls for a comprehensive and systematic review to better summarizes and guide this promising domain. In this survey paper, first, we propose a systematic taxonomy that categorizes the LLM domain-specialization techniques based on the accessibility to LLMs and summarizes the framework for all the subcategories as well as their relations and differences to each other. We also present a comprehensive taxonomy of critical application domains that can benefit from specialized LLMs, discussing their practical significance and open challenges. Furthermore, we offer insights into the current research status and future trends in this area.
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.
Residual networks (ResNets) have displayed impressive results in pattern recognition and, recently, have garnered considerable theoretical interest due to a perceived link with neural ordinary differential equations (neural ODEs). This link relies on the convergence of network weights to a smooth function as the number of layers increases. We investigate the properties of weights trained by stochastic gradient descent and their scaling with network depth through detailed numerical experiments. We observe the existence of scaling regimes markedly different from those assumed in neural ODE literature. Depending on certain features of the network architecture, such as the smoothness of the activation function, one may obtain an alternative ODE limit, a stochastic differential equation or neither of these. These findings cast doubts on the validity of the neural ODE model as an adequate asymptotic description of deep ResNets and point to an alternative class of differential equations as a better description of the deep network limit.
Graph neural networks (GNNs) are a popular class of machine learning models whose major advantage is their ability to incorporate a sparse and discrete dependency structure between data points. Unfortunately, GNNs can only be used when such a graph-structure is available. In practice, however, real-world graphs are often noisy and incomplete or might not be available at all. With this work, we propose to jointly learn the graph structure and the parameters of graph convolutional networks (GCNs) by approximately solving a bilevel program that learns a discrete probability distribution on the edges of the graph. This allows one to apply GCNs not only in scenarios where the given graph is incomplete or corrupted but also in those where a graph is not available. We conduct a series of experiments that analyze the behavior of the proposed method and demonstrate that it outperforms related methods by a significant margin.
Detecting carried objects is one of the requirements for developing systems to reason about activities involving people and objects. We present an approach to detect carried objects from a single video frame with a novel method that incorporates features from multiple scales. Initially, a foreground mask in a video frame is segmented into multi-scale superpixels. Then the human-like regions in the segmented area are identified by matching a set of extracted features from superpixels against learned features in a codebook. A carried object probability map is generated using the complement of the matching probabilities of superpixels to human-like regions and background information. A group of superpixels with high carried object probability and strong edge support is then merged to obtain the shape of the carried object. We applied our method to two challenging datasets, and results show that our method is competitive with or better than the state-of-the-art.
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