Fourier transform of a power of a Euclidean distance, i.e., According to Jensen's inequality and Lipschitzness assumption, we have X According to the law of total probability and Theorem 4.1, we have P { Y A feasible solution to Equation (1) can be quickly found. Pseudocode for Algorithm 2 The pseudocode for the constrained optimization is detailed in Algorithm 2. 18 Algorithm 2 Robust Optimization Method with EOC Constraint Input: Initiate Array A of shape 2 A M that stores the max possible step. Our proposed algorithm is highly computationally efficient.

In this work, we give a ${\rm poly}(d,k)$ time and sample algorithm for efficiently learning the parameters of a mixture of $k$ spherical distributions in $d$ dimensions. Unlike all previous methods, our techniques apply to heavy-tailed distributions and include examples that do not even have finite covariances. Our method succeeds whenever the cluster distributions have a characteristic function with sufficiently heavy tails. Such distributions include the Laplace distribution but crucially exclude Gaussians. All previous methods for learning mixture models relied implicitly or explicitly on the low-degree moments. Even for the case of Laplace distributions, we prove that any such algorithm must use super-polynomially many samples. Our method thus adds to the short list of techniques that bypass the limitations of the method of moments. Somewhat surprisingly, our algorithm does not require any minimum separation between the cluster means. This is in stark contrast to spherical Gaussian mixtures where a minimum $\ell_2$-separation is provably necessary even information-theoretically [Regev and Vijayaraghavan '17]. Our methods compose well with existing techniques and allow obtaining ''best of both worlds" guarantees for mixtures where every component either has a heavy-tailed characteristic function or has a sub-Gaussian tail with a light-tailed characteristic function. Our algorithm is based on a new approach to learning mixture models via efficient high-dimensional sparse Fourier transforms. We believe that this method will find more applications to statistical estimation. As an example, we give an algorithm for consistent robust mean estimation against noise-oblivious adversaries, a model practically motivated by the literature on multiple hypothesis testing. It was formally proposed in a recent Master's thesis by one of the authors, and has already inspired follow-up works.

Generating high-fidelity time series data using generative adversarial networks (GANs) remains a challenging task, as it is difficult to capture the temporal dependence of joint probability distributions induced by time-series data. Towards this goal, a key step is the development of an effective discriminator to distinguish between time series distributions. We propose the so-called PCF-GAN, a novel GAN that incorporates the path characteristic function (PCF) as the principled representation of time series distribution into the discriminator to enhance its generative performance. On the one hand, we establish theoretical foundations of the PCF distance by proving its characteristicity, boundedness, differentiability with respect to generator parameters, and weak continuity, which ensure the stability and feasibility of training the PCF-GAN. On the other hand, we design efficient initialisation and optimisation schemes for PCFs to strengthen the discriminative power and accelerate training efficiency. To further boost the capabilities of complex time series generation, we integrate the auto-encoder structure via sequential embedding into the PCF-GAN, which provides additional reconstruction functionality. Extensive numerical experiments on various datasets demonstrate the consistently superior performance of PCF-GAN over state-of-the-art baselines, in both generation and reconstruction quality.

Can we predict how well a team of individuals will perform together? How should individuals be rewarded for their contributions to the team performance? Cooperative game theory gives us a powerful set of tools for answering these questions: the characteristic function and solution concepts like the Shapley Value. There are two major difficulties in applying these techniques to real world problems: first, the characteristic function is rarely given to us and needs to be learned from data. Second, the Shapley Value is combinatorial in nature. We introduce a parametric model called cooperative game abstractions (CGAs) for estimating characteristic functions from data. CGAs are easy to learn, readily interpretable, and crucially allows linear-time computation of the Shapley Value. We provide identification results and sample complexity bounds for CGA models as well as error bounds in the estimation of the Shapley Value using CGAs. We apply our methods to study teams of artificial RL agents as well as real world teams from professional sports.

Inspired by graph-based methodologies, we introduce a novel graph-spanning algorithm designed to identify changes in both offline and online data across low to high dimensions. This versatile approach is applicable to Euclidean and graph-structured data with unknown distributions, while maintaining control over error probabilities. Theoretically, we demonstrate that the algorithm achieves high detection power when the magnitude of the change surpasses the lower bound of the minimax separation rate, which scales on the order of $\sqrt{nd}$. Our method outperforms other techniques in terms of accuracy for both Gaussian and non-Gaussian data. Notably, it maintains strong detection power even with small observation windows, making it particularly effective for online environments where timely and precise change detection is critical.

In this contribution, we focus on invariances to symmetric additive noise on each of the data generating distributions.

Nonparametric two-sample tests based on RKHS distances (Eric et al., 2008; Fromont et al., 2012;