We propose an automatic approach for manifold nonlinear autoregressive with exogenous inputs (mNARX) modeling that leverages the feature-based structure of functional-NARX (F-NARX) modeling. This novel approach, termed mNARX+, preserves the key strength of the mNARX framework, which is its expressivity allowing it to model complex dynamical systems, while simultaneously addressing a key limitation: the heavy reliance on domain expertise to identify relevant auxiliary quantities and their causal ordering. Our method employs a data-driven, recursive algorithm that automates the construction of the mNARX model sequence. It operates by sequentially selecting temporal features based on their correlation with the model prediction residuals, thereby automatically identifying the most critical auxiliary quantities and the order in which they should be modeled. This procedure significantly reduces the need for prior system knowledge. We demonstrate the effectiveness of the mNARX+ algorithm on two case studies: a Bouc-Wen oscillator with strong hysteresis and a complex aero-servo-elastic wind turbine simulator. The results show that the algorithm provides a systematic, data-driven method for creating accurate and stable surrogate models for complex dynamical systems.

This TV-regularized convex program on the space of measures allows to recover a sparse measure using a noisy observation from an appropriate measurement operator. While previous works have uncovered the central role played by this operator and its associated kernel in order to get estimation error bounds, the latter requires a technical local positive curvature (LPC) assumption to be verified on a case-by-case basis. In practice, this yields only few LPC-kernels for which this condition is proved. At the heart of our contribution lies the kernel switch, which uncouples the model kernel from the LPC assumption: it enables to leverage any known LPC-kernel as a pivot kernel to prove error bounds, provided embedding conditions are verified between the model and pivot RKHS. We increment the list of LPC-kernels, proving that the "sinc-4" kernel, used for signal recovery and mixture problems, does satisfy the LPC assumption. Furthermore, we also show that the BLASSO localisation error around the true support decreases with the noise level, leading to effective near regions. This improves on known results where this error is fixed with some parameters depending on the model kernel. We illustrate the interest of our results in the case of translation-invariant mixture model estimation, using bandlimiting smoothing and sketching techniques to reduce the computational burden of BLASSO.

We propose a novel particle-based variational inference method designed to work with multimodal distributions. Our approach, referred to as Branched Stein Variational Gradient Descent (BSVGD), extends the classical Stein Variational Gradient Descent (SVGD) algorithm by incorporating a random branching mechanism that encourages the exploration of the state space. In this work, a theoretical guarantee for the convergence in distribution is presented, as well as numerical experiments to validate the suitability of our algorithm. Performance comparisons between the BSVGD and the SVGD are presented using the Wasserstein distance between samples and the corresponding computational times.

While average treatment effects (ATE) and conditional average treatment effects (CATE) provide valuable population- and subgroup-level summaries, they fail to capture uncertainty at the individual level. For high-stakes decision-making, individual treatment effect (ITE) estimates must be accompanied by valid prediction intervals that reflect heterogeneity and unit-specific uncertainty. However, the fundamental unidentifiability of ITEs limits the ability to derive precise and reliable individual-level uncertainty estimates. To address this challenge, we investigate the role of a cross-world correlation parameter, $ ρ(x) = cor(Y(1), Y(0) | X = x) $, which describes the dependence between potential outcomes, given covariates, in the Neyman-Rubin super-population model with i.i.d. units. Although $ ρ$ is fundamentally unidentifiable, we argue that in most real-world applications, it is possible to impose reasonable and interpretable bounds informed by domain-expert knowledge. Given $ρ$, we design prediction intervals for ITE, achieving more stable and accurate coverage with substantially shorter widths; often less than 1/3 of those from competing methods. The resulting intervals satisfy coverage guarantees $P\big(Y(1) - Y(0) \in C_{ITE}(X)\big) \geq 1 - α$ and are asymptotically optimal under Gaussian assumptions. We provide strong theoretical and empirical arguments that cross-world assumptions can make individual uncertainty quantification both practically informative and statistically valid.

The Sliced Gromov-Wasserstein (SGW) distance, aiming to relieve the computational cost of solving a non-convex quadratic program that is the Gromov-Wasserstein distance, utilizes projecting directions sampled uniformly from unit hyperspheres. This slicing mechanism incurs unnecessary computational costs due to uninformative directions, which also affects the representative power of the distance. However, finding a more appropriate distribution over the projecting directions (slicing distribution) is often an optimization problem in itself that comes with its own computational cost. In addition, with more intricate distributions, the sampling itself may be expensive. As a remedy, we propose an optimization-free slicing distribution that provides fast sampling for the Monte Carlo approximation. We do so by introducing the Relation-Aware Projecting Direction (RAPD), effectively capturing the pairwise association of each of two pairs of random vectors, each following their ambient law. This enables us to derive the Relation-Aware Slicing Distribution (RASD), a location-scale law corresponding to sampled RAPDs. Finally, we introduce the RASGW distance and its variants, e.g., IWRASGW (Importance Weighted RASGW), which overcome the shortcomings experienced by SGW. We theoretically analyze its properties and substantiate its empirical prowess using extensive experiments on various alignment tasks.

In recent years, contrastive learning has achieved state-of-the-art performance in the territory of self-supervised representation learning. Many previous works have attempted to provide the theoretical understanding underlying the success of contrastive learning. Almost all of them rely on a default assumption, i.e., the label consistency assumption, which may not hold in practice (the probability of failure is called labeling error) due to the strength and randomness of common augmentation strategies, such as random resized crop (RRC). This paper investigates the theoretical impact of labeling error on the downstream classification performance of contrastive learning. We first reveal several significant negative impacts of labeling error on downstream classification risk. To mitigate these impacts, data dimensionality reduction method (e.g., singular value decomposition, SVD) is applied on original data to reduce false positive samples, and establish both theoretical and empirical evaluations. Moreover, it is also found that SVD acts as a double-edged sword, which may lead to the deterioration of downstream classification accuracy due to the reduced connectivity of the augmentation graph. Based on the above observations, we give the augmentation suggestion that we should use some moderate embedding dimension (such as $512, 1024$ in our experiments), data inflation, weak augmentation, and SVD to ensure large graph connectivity and small labeling error to improve model performance.

We study PAC and online learnability of hypothesis classes formed by copies of a countably infinite graph G, where each copy is induced by permuting G's vertices. This corresponds to learning a graph's labeling, knowing its structure and label set. We consider classes where permutations move only finitely many vertices. Our main result shows that PAC learnability of all such finite-support copies implies online learnability of the full isomorphism type of G, and is equivalent to the condition of automorphic triviality. We also characterize graphs where copies induced by swapping two vertices are not learnable, using a relaxation of the extension property of the infinite random graph. Finally, we show that, for all G and k>2, learnability for k-vertex permutations is equivalent to that for 2-vertex permutations, yielding a four-class partition of infinite graphs, whose complexity we also determine using tools coming from both descriptive set theory and computability theory.

Symbolic regression is the machine learning method for learning functions from data. After a brief overview of the symbolic regression landscape, I will describe the two main challenges that traditional algorithms face: they have an unknown (and likely significant) probability of failing to find any given good function, and they suffer from ambiguity and poorly-justified assumptions in their function-selection procedure. To address these I propose an exhaustive search and model selection by the minimum description length principle, which allows accuracy and complexity to be directly traded off by measuring each in units of information. I showcase the resulting publicly available Exhaustive Symbolic Regression algorithm on three open problems in astrophysics: the expansion history of the universe, the effective behaviour of gravity in galaxies and the potential of the inflaton field. In each case the algorithm identifies many functions superior to the literature standards. This general purpose methodology should find widespread utility in science and beyond.

Foundation multi-modal models are often designed by stitching of multiple existing pretrained uni-modal models: for example, an image classifier with an text model. This stitching process is performed by training a connector module that aims to align the representation spaces of these uni-modal models towards a multi-modal objective. However, given the complexity of training such connectors on large scale web-based datasets coupled with the ever-increasing number of available pretrained uni-modal models, the task of uni-modal models selection and subsequent connector module training becomes computationally demanding. To address this under-studied critical problem, we propose Hypernetwork Model Alignment (Hyma), a novel all-in-one solution for optimal uni-modal model selection and connector training by leveraging hypernetworks. Specifically, our framework utilizes the parameter prediction capability of a hypernetwork to obtain jointly trained connector modules for $N \times M$ combinations of uni-modal models. In our experiments, Hyma reduces the cost of searching for the best performing uni-modal model pair by $10\times$, while matching the ranking and trained connector performance obtained via grid search across a suite of diverse multi-modal benchmarks.

Characterizing the convergence of real-valued or vector-v alued sequences is a key theoretical problem in data science, where the sequence index typically correspon ds to the number of iterations of an iterative algorithm (such as in optimization and signal processing) o r the number of observations (as in statistics and machine learning). This characterization can be done in mostly two ways, asymptotically or non-asymptotically. In an asymptotic analysis, an asymptotic e quivalent of the sequence is identified, which readily allows comparisons with other algorithms; however, without further analysis, the behavior at any finite time cannot be controlled. This is exactly what non-as ymptotic analysis aims to achieve, by providing bounds that are valid even for a finite index, but then only pro viding bounds that cannot always be compared. While the two approaches have their own merits, in this paper, we focus on asymptotic analysis and sequences that tend to their limit at a sub-exponential r ate that is a power of the sequence index. The main goal of this paper is to show how a classical tool from signal processing, control theory, and electrical engineering ( Oppenheim et al., 1996), the z -transform method ( Jury, 1964), can be used in this context with a striking efficiency at obtaining asymptotic eq uivalents for the class of algorithms that can be seen as iterations of potentially random linear operators i n a Hilbert space. This includes gradient descent for quadratic optimization problems as well as its accelera ted and stochastic variants ( Nesterov, 2018), 1 Landweber iterations in inverse problems ( Benning and Burger, 2018), or gossip algorithms in distributed computing ( Boyd et al., 2006).