We operate through the lens of ordinary differential equations and control theory to study the concept of observability in the context of neural state-space models and the Mamba architecture. We develop strategies to enforce observability, which are tailored to a learning context, specifically where the hidden states are learnable at initial time, in conjunction to over its continuum, and high-dimensional. We also highlight our methods emphasize eigenvalues, roots of unity, or both. Our methods effectuate computational efficiency when enforcing observability, sometimes at great scale. We formulate observability conditions in machine learning based on classical control theory and discuss their computational complexity. Our nontrivial results are fivefold. We discuss observability through the use of permutations in neural applications with learnable matrices without high precision. We present two results built upon the Fourier transform that effect observability with high probability up to the randomness in the learning. These results are worked with the interplay of representations in Fourier space and their eigenstructure, nonlinear mappings, and the observability matrix. We present a result for Mamba that is similar to a Hautus-type condition, but instead employs an argument using a Vandermonde matrix instead of eigenvectors. Our final result is a shared-parameter construction of the Mamba system, which is computationally efficient in high exponentiation. We develop a training algorithm with this coupling, showing it satisfies a Robbins-Monro condition under certain orthogonality, while a more classical training procedure fails to satisfy a contraction with high Lipschitz constant.

Understanding the inner workings of neural embeddings, particularly in models such as BERT, remains a challenge because of their high-dimensional and opaque nature. This paper proposes a framework for uncovering the specific dimensions of vector embeddings that encode distinct linguistic properties (LPs). We introduce the Linguistically Distinct Sentence Pairs (LDSP-10) dataset, which isolates ten key linguistic features such as synonymy, negation, tense, and quantity. Using this dataset, we analyze BERT embeddings with various methods, including the Wilcoxon signed-rank test, mutual information, and recursive feature elimination, to identify the most influential dimensions for each LP. We introduce a new metric, the Embedding Dimension Impact (EDI) score, which quantifies the relevance of each embedding dimension to a LP. Our findings show that certain properties, such as negation and polarity, are robustly encoded in specific dimensions, while others, like synonymy, exhibit more complex patterns. This study provides insights into the interpretability of embeddings, which can guide the development of more transparent and optimized language models, with implications for model bias mitigation and the responsible deployment of AI systems.

A fundamental challenge in the cognitive sciences is discovering the dynamics that govern behaviour. Take the example of spoken language, which is characterised by a highly variable and complex set of physical movements that map onto the small set of cognitive units that comprise language. What are the fundamental dynamical principles behind the movements that structure speech production? In this study, we discover models in the form of symbolic equations that govern articulatory gestures during speech. A sparse symbolic regression algorithm is used to discover models from kinematic data on the tongue and lips. We explore these candidate models using analytical techniques and numerical simulations, and find that a second-order linear model achieves high levels of accuracy, but a nonlinear force is required to properly model articulatory dynamics in approximately one third of cases. This supports the proposal that an autonomous, nonlinear, second-order differential equation is a viable dynamical law for articulatory gestures in speech. We conclude by identifying future opportunities and obstacles in data-driven model discovery and outline prospects for discovering the dynamical principles that govern language, brain and behaviour.

In the near future, Structural Health Monitoring (SHM) technologies will be capable of overcoming the drawbacks in the current maintenance and life-cycle management paradigms, namely: cost, increased downtime, less-than-optimal safety management paradigm and the limited applicability of fully-autonomous operations. In the context of SHM, one of the most challenging tasks is damage quantification. Current methods face accuracy and/or robustness issues when it comes to varying operating and environmental conditions. In addition, the damage/no-damage paradigm of current frameworks does not offer much information to maintainers on the ground for proper decision-making. In this study, a novel structural damage quantification framework is proposed based on widely-used Damage Indices (DIs) and Gaussian Process Regression Models (GPRMs). The novelty lies in calculating the probability of an incoming test DI point originating from a specific state, which allows for probability-educated decision-making. This framework is applied to three test cases: a Carbon Fiber-Reinforced Plastic (CFRP) coupon with attached weights as simulated damage, an aluminum coupon with a notch, and an aluminum coupon with attached weights as simulated damage under varying loading states. The state prediction method presented herein is applied to single-state quantification in the first two test cases, as well as the third one assuming the loading state is known. Finally, the proposed method is applied to the third test case assuming neither the damage size nor the load is known in order to predict both simultaneously from incoming DI test points. In applying this framework, two forms of GPRMs (standard and variational heteroscedastic) are used in order to critically assess their performances with respect to the three test cases.

Next Generation Reservoir Computing (NGRC) is a low-cost machine learning method for forecasting chaotic time series from data. However, ensuring the dynamical stability of NGRC models during autonomous prediction remains a challenge. In this work, we uncover a key connection between the numerical conditioning of the NGRC feature matrix -- formed by polynomial evaluations on time-delay coordinates -- and the long-term NGRC dynamics. Merging tools from numerical linear algebra and ergodic theory of dynamical systems, we systematically study how the feature matrix conditioning varies across hyperparameters. We demonstrate that the NGRC feature matrix tends to be ill-conditioned for short time lags and high-degree polynomials. Ill-conditioning amplifies sensitivity to training data perturbations, which can produce unstable NGRC dynamics. We evaluate the impact of different numerical algorithms (Cholesky, SVD, and LU) for solving the regularized least-squares problem.

Bootstrap is a common tool for quantifying uncertainty in data analysis. However, besides additional computational costs in the application of the bootstrap on massive data, a challenging problem in bootstrap based inference under Differential Privacy consists in the fact that it requires repeated access to the data. As a consequence, bootstrap based differentially private inference requires a significant increase of the privacy budget, which on the other hand comes with a substantial loss in statistical accuracy. A potential solution to reconcile the conflicting goals of statistical accuracy and privacy is to analyze the data under parametric model assumptions and in the last decade, several parametric bootstrap methods for inference under privacy have been investigated. However, uncertainty quantification by parametric bootstrap is only valid if the the quantities of interest can be identified as the parameters of a statistical model and the imposed model assumptions are (at least approximately) satisfied. An alternative to parametric methods is the empirical bootstrap that is a widely used tool for non-parametric inference and well studied in the non-private regime. However, under privacy, less insight is available. In this paper, we propose a private empirical $m$ out of $n$ bootstrap and validate its consistency and privacy guarantees under Gaussian Differential Privacy. Compared to the the private $n$ out of $n$ bootstrap, our approach has several advantages. First, it comes with less computational costs, in particular for massive data. Second, the proposed procedure needs less additional noise in the bootstrap iterations, which leads to an improved statistical accuracy while asymptotically guaranteeing the same level of privacy. Third, we demonstrate much better finite sample properties compared to the currently available procedures.

In particular, Theorem 5 applies when null A δ = { δ }, i.e., when θ null A is just a standard, non-robust, solution. This is different from [56], whose main result is only applicable to solutions satisfying the infinitely many constraints f (θ, δ) 0, δ A δ i, i = 1,...,N, where A δ i is tuned to the Wasserstein bound. As previously noted, R plays the role of a tunable parameter, and the result in Theorem 5 holds for any choice of the value ofR . As a consequence, the user can play with R to optimize the bound on Risk ( θ null A) given in Theorem 5. As R increases, s A, null A (and, thereby, ε (s A, null A)) tends to increase while µ/R diminishes. While the best compromise is difficult to foresee, one can experimentally try various choices R 1 < R 2 < < R i < R h and select the one giving the best result. The corresponding confidence level can be bounded as follows: P Nnull D: Risk (θ null A) > ε (s A, null A,i) + µ R i for at least one i { 1,...h } null h null i =1P Nnull D: Risk (θ null A) > ε (s A, null A,i) + µ R i null h null i =1β = hβ, 29 from which P Nnull D: Risk ( θ null A) ε ( s A, null A,i) + µ R i for all i = 1,...h null 1 hβ.

Selecting high-quality candidates from large datasets is critical in applications such as drug discovery, precision medicine, and alignment of large language models (LLMs). While Conformal Selection (CS) provides rigorous uncertainty quantification, it is limited to univariate responses and scalar criteria. To address this issue, we propose Multivariate Conformal Selection (mCS), a generalization of CS designed for multivariate response settings. Our method introduces regional monotonicity and employs multivariate nonconformity scores to construct conformal p-values, enabling finite-sample False Discovery Rate (FDR) control. We present two variants: mCS-dist, using distance-based scores, and mCS-learn, which learns optimal scores via differentiable optimization. Experiments on simulated and real-world datasets demonstrate that mCS significantly improves selection power while maintaining FDR control, establishing it as a robust framework for multivariate selection tasks.

A clustered adaptive intervention (cAI) is a pre-specified sequence of decision rules that guides practitioners on how best - and based on which measures - to tailor cluster-level intervention to improve outcomes at the level of individuals within the clusters. A clustered sequential multiple assignment randomized trial (cSMART) is a type of trial that is used to inform the empirical development of a cAI. The most common type of secondary aim in a cSMART focuses on assessing causal effect moderation by candidate tailoring variables. We introduce a clustered Q-learning framework with the M-out-of-N Cluster Bootstrap using data from a cSMART to evaluate whether a set of candidate tailoring variables may be useful in defining an optimal cAI. This approach could construct confidence intervals (CI) with near-nominal coverage to assess parameters indexing the causal effect moderation function. Specifically, it allows reliable inferences concerning the utility of candidate tailoring variables in constructing a cAI that maximizes a mean end-of-study outcome even when "non-regularity", a well-known challenge exists. Simulations demonstrate the numerical performance of the proposed method across varying non-regularity conditions and investigate the impact of varying number of clusters and intra-cluster correlation coefficient on CI coverage. Methods are applied on ADEPT dataset to inform the construction of a clinic-level cAI for improving evidence-based practice in treating mood disorders.

Many learning problems involve multiple patterns and varying degrees of uncertainty dependent on the covariates. Advances in Deep Learning (DL) have addressed these issues by learning highly nonlinear input-output dependencies. However, model interpretability and Uncertainty Quantification (UQ) have often straggled behind. In this context, we introduce the Competitive/Collaborative Fusion of Experts (CoCoAFusE), a novel, Bayesian Covariates-Dependent Modeling technique. CoCoAFusE builds on the very philosophy behind Mixtures of Experts (MoEs), blending predictions from several simple sub-models (or "experts") to achieve high levels of expressiveness while retaining a substantial degree of local interpretability. Our formulation extends that of a classical Mixture of Experts by contemplating the fusion of the experts' distributions in addition to their more usual mixing (i.e., superimposition). Through this additional feature, CoCoAFusE better accommodates different scenarios for the intermediate behavior between generating mechanisms, resulting in tighter credible bounds on the response variable. Indeed, only resorting to mixing, as in classical MoEs, may lead to multimodality artifacts, especially over smooth transitions. Instead, CoCoAFusE can avoid these artifacts even under the same structure and priors for the experts, leading to greater expressiveness and flexibility in modeling. This new approach is showcased extensively on a suite of motivating numerical examples and a collection of real-data ones, demonstrating its efficacy in tackling complex regression problems where uncertainty is a key quantity of interest.