Data-driven spectral analysis of Koopman operators is a powerful tool for understanding numerous real-world dynamical systems, from neuronal activity to variations in sea surface temperature. The Koopman operator acts on a function space and is most commonly studied on the space of square-integrable functions. However, defining it on a suitable reproducing kernel Hilbert space (RKHS) offers numerous practical advantages, including pointwise predictions with error bounds, improved spectral properties that facilitate computations, and more efficient algorithms, particularly in high dimensions. We introduce the first general, provably convergent, data-driven algorithms for computing spectral properties of Koopman and Perron--Frobenius operators on RKHSs. These methods efficiently compute spectra and pseudospectra with error control and spectral measures while exploiting the RKHS structure to avoid the large-data limits required in the $L^2$ settings. The function space is determined by a user-specified kernel, eliminating the need for quadrature-based sampling as in $L^2$ and enabling greater flexibility with finite, externally provided datasets. Using the Solvability Complexity Index hierarchy, we construct adversarial dynamical systems for these problems to show that no algorithm can succeed in fewer limits, thereby proving the optimality of our algorithms. Notably, this impossibility extends to randomized algorithms and datasets. We demonstrate the effectiveness of our algorithms on challenging, high-dimensional datasets arising from real-world measurements and high-fidelity numerical simulations, including turbulent channel flow, molecular dynamics of a binding protein, Antarctic sea ice concentration, and Northern Hemisphere sea surface height. The algorithms are publicly available in the software package $\texttt{SpecRKHS}$.

Common workflows in machine learning and statistics rely on the ability to partition the information in a data set into independent portions. Recent work has shown that this may be possible even when conventional sample splitting is not (e.g., when the number of samples $n=1$, or when observations are not independent and identically distributed). However, the approaches that are currently available to decompose multivariate Gaussian data require knowledge of the covariance matrix. In many important problems (such as in spatial or longitudinal data analysis, and graphical modeling), the covariance matrix may be unknown and even of primary interest. Thus, in this work we develop new approaches to decompose Gaussians with unknown covariance. First, we present a general algorithm that encompasses all previous decomposition approaches for Gaussian data as special cases, and can further handle the case of an unknown covariance. It yields a new and more flexible alternative to sample splitting when $n>1$. When $n=1$, we prove that it is impossible to partition the information in a multivariate Gaussian into independent portions without knowing the covariance matrix. Thus, we use the general algorithm to decompose a single multivariate Gaussian with unknown covariance into dependent parts with tractable conditional distributions, and demonstrate their use for inference and validation. The proposed decomposition strategy extends naturally to Gaussian processes. In simulation and on electroencephalography data, we apply these decompositions to the tasks of model selection and post-selection inference in settings where alternative strategies are unavailable.

In recent years, the integration of Automated Planning (AP) and Reinforcement Learning (RL) has seen a surge of interest. To perform this integration, a general framework for Sequential Decision Making (SDM) would prove immensely useful, as it would help us understand how AP and RL fit together. In this preliminary work, we attempt to provide such a framework, suitable for any method ranging from Classical Planning to Deep RL, by drawing on concepts from Probability Theory and Bayesian inference. We formulate an SDM task as a set of training and test Markov Decision Processes (MDPs), to account for generalization. We provide a general algorithm for SDM which we hypothesize every SDM method is based on. According to it, every SDM algorithm can be seen as a procedure that iteratively improves its solution estimate by leveraging the task knowledge available. Finally, we derive a set of formulas and algorithms for calculating interesting properties of SDM tasks and methods, which make possible their empirical evaluation and comparison.

During the past decade, deep learning has transformed a number of historically challenging problems in computer vision, natural language processing, game intelligence, etc. In many of these applications, the trained neural networks used to solve these problems are over-parameterized. That is, the neural networks have far more parameters than the number of data points used for training. In this setting, a neural network can typically fit any training data - including random labels [95] - making it hard to explain why deep learning methods generalize so well [36]. Moreover, the practical performance of neural networks often improves as the number of parameters grow [55,84]. These observations have led to the study of the potential implicit regularization (sometimes called implicit bias) imposed by the gradient based methods and different network architectures [8, 68, 69]. It may seem surprising that there is a link to generalized hardness of approximation (GHA), as this phenomenon - at a first glance - may seem disconnected from implicit regularization. However, the GHA phenomenon (see 1.2), which first appeared in [13] (see also [2] Chapter 8) and analyzed [13, 34, 41] in connection with robust and convex optimization [20, 21, 63, 64], typically stem from regularization problems (e.g.

The unprecedented success of deep learning (DL) makes it unchallenged when it comes to classification problems. However, it is well established that the current DL methodology produces universally unstable neural networks (NNs). The instability problem has caused an enormous research effort -- with a vast literature on so-called adversarial attacks -- yet there has been no solution to the problem. Our paper addresses why there has been no solution to the problem, as we prove the following mathematical paradox: any training procedure based on training neural networks for classification problems with a fixed architecture will yield neural networks that are either inaccurate or unstable (if accurate) -- despite the provable existence of both accurate and stable neural networks for the same classification problems. The key is that the stable and accurate neural networks must have variable dimensions depending on the input, in particular, variable dimensions is a necessary condition for stability. Our result points towards the paradox that accurate and stable neural networks exist, however, modern algorithms do not compute them. This yields the question: if the existence of neural networks with desirable properties can be proven, can one also find algorithms that compute them? There are cases in mathematics where provable existence implies computability, but will this be the case for neural networks? The contrary is true, as we demonstrate how neural networks can provably exist as approximate minimisers to standard optimisation problems with standard cost functions, however, no randomised algorithm can compute them with probability better than 1/2.

Byron Reese: This is "Voices in AI", brought to you by Gigaom. Today our guest is Gregory Piatetsky. Twenty years ago, he founded and continues to operate a site called KDnuggets about knowledge discovery. It's dedicated to the various topics he's interested in. Many people think it's a must-read resource. It has over 400,000 regular monthly readers. He holds an MS and a PhD in computer science from NYU. Gregory Piatetsky: Thank you, Byron. Glad to be with you. I always like to start off with definitions, because in a way we're in such a nascent field in the grand scheme of things that people don't necessarily start off agreeing on what terms mean. How do you define artificial intelligence? Artificial intelligence is really machines doing things that people think require intelligence, and by that definition the goalposts of artificial intelligence are constantly moving. It was considered very intelligent to play checkers back in the 1950s, then there was a program. The next boundary was playing chess, and then computers mastered it.

Generalizing empirical findings to new environments, settings, or populations is essential in most scientific explorations. This article treats a particular problem of generalizability, called "transportability", defined as a license to transfer information learned in experimental studies to a different population, on which only observational studies can be conducted. Given a set of assumptions concerning commonalities and differences between the two populations, Pearl and Bareinboim (2011) derived sufficient conditions that permit such transfer to take place. This article summarizes their findings and supplements them with an effective procedure for deciding when and how transportability is feasible. It establishes a necessary and sufficient condition for deciding when causal effects in the target population are estimable from both the statistical information available and the causal information transferred from the experiments. The article further provides a complete algorithm for computing the transport formula, that is, a way of combining observational and experimental information to synthesize bias-free estimate of the desired causal relation. Finally, the article examines the differences between transportability and other variants of generalizability.

We introduce a general family of kernels based on weighted transducers or rational relations, rational kernels, that can be used for analysis of variable-length sequences or more generally weighted automata, in applications such as computational biology or speech recognition. We show that rational kernels can be computed efficiently using a general algorithm of composition of weighted transducers and a general single-source shortest-distance algorithm. We also describe several general families of positive definite symmetric rational kernels. These general kernels can be combined with Support Vector Machines to form efficient and powerful techniques for spoken-dialog classification: highly complex kernels become easy to design and implement and lead to substantial improvements in the classification accuracy. We also show that the string kernels considered in applications to computational biology are all specific instances of rational kernels.

We introduce a general family of kernels based on weighted transducers or rational relations, rational kernels, that can be used for analysis of variable-length sequences or more generally weighted automata, in applications such as computational biology or speech recognition. We show that rational kernels can be computed efficiently using a general algorithm of composition of weighted transducers and a general single-source shortest-distance algorithm. We also describe several general families of positive definite symmetric rational kernels. These general kernels can be combined with Support Vector Machines to form efficient and powerful techniques for spoken-dialog classification: highly complex kernels become easy to design and implement and lead to substantial improvements in the classification accuracy. We also show that the string kernels considered in applications to computational biology are all specific instances of rational kernels.

We introduce a general family of kernels based on weighted transducers orrational relations, rational kernels, that can be used for analysis of variable-length sequences or more generally weighted automata, in applications suchas computational biology or speech recognition. We show that rational kernels can be computed efficiently using a general algorithm ofcomposition of weighted transducers and a general single-source shortest-distance algorithm. We also describe several general families of positive definite symmetric rational kernels. These general kernels can be combined with Support Vector Machines to form efficient and powerful techniquesfor spoken-dialog classification: highly complex kernels become easy to design and implement and lead to substantial improvements inthe classification accuracy. We also show that the string kernels considered in applications to computational biology are all specific instances ofrational kernels.