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In this paper we fully describe the trajectory of gradient flow over diagonal linear networks in the limit of vanishing initialisation. We show that the limiting flow successively jumps from a saddle of the training loss to another until reaching the minimum $\ell_1$-norm solution. This saddle-to-saddle dynamics translates to an incremental learning process as each saddle corresponds to the minimiser of the loss constrained to an active set outside of which the coordinates must be zero. We explicitly characterise the visited saddles as well as the jumping times through a recursive algorithm reminiscent of the LARS algorithm used for computing the Lasso path. Our proof leverages a convenient arc-length time-reparametrisation which enables to keep track of the heteroclinic transitions between the jumps. Our analysis requires negligible assumptions on the data, applies to both under and overparametrised settings and covers complex cases where there is no monotonicity of the number of active coordinates. We provide numerical experiments to support our findings.

This paper presents novel methodologies for conducting practical differentially private (DP) estimation and inference in high-dimensional linear regression. We start by proposing a differentially private Bayesian Information Criterion (BIC) for selecting the unknown sparsity parameter in DP-Lasso, eliminating the need for prior knowledge of model sparsity, a requisite in the existing literature. Then we propose a differentially private debiased LASSO algorithm that enables privacy-preserving inference on regression parameters. Our proposed method enables accurate and private inference on the regression parameters by leveraging the inherent sparsity of high-dimensional linear regression models. Additionally, we address the issue of multiple testing in high-dimensional linear regression by introducing a differentially private multiple testing procedure that controls the false discovery rate (FDR). This allows for accurate and privacy-preserving identification of significant predictors in the regression model. Through extensive simulations and real data analysis, we demonstrate the efficacy of our proposed methods in conducting inference for high-dimensional linear models while safeguarding privacy and controlling the FDR.

This paper develops a novel minimal-state operational semantics for higher-order functional languages which uses only the call stack and two source program points as the complete state information: there is no environment, no substitution, no continuation, etc. We prove this form of operational semantics is equivalent to standard presentations. We then show how this approach can open the door to potential new applications: we define a program analysis as a direct finitization of this operational semantics. The program analysis that naturally emerges has a number of novel and interesting properties compared to standard program analyses for higher-order programs: for example, it can infer recurrences, and does not need value widening. We both give a formal definition of the analysis and describe our current implementation.

Shape optimization with respect to eigenvalues of a cavity plays an important role in the design of new resonators or in the optimization of existing ones. In our paper, we propose a gradient-based optimization scheme, which we enhance with closed-form shape derivatives of the system matrices. Based on these, we can compute accurate derivatives of eigenvalues, eigenmodes and the cost function with respect to the geometry, which significantly reduces the computational effort of the optimizer. We demonstrate our work by applying it to the 9-cell TESLA cavity, for which we tune the design parameters of the computational model to match the design criteria for devices in realistic use cases. Since eigenvalues may cross during the shape optimization of a cavity, we propose a new algorithm based on an eigenvalue matching procedure, to ensure the optimization of the desired mode in order to also enable successful matching along large shape variations.

This paper introduces a crowd modeling and motion control approach that employs diffusion adaptation within an adaptive network. In the network, nodes collaboratively address specific estimation problems while simultaneously moving as agents governed by certain motion control mechanisms. Our research delves into the behaviors of agents when they encounter spatial constraints. Within this framework, agents pursue several objectives, such as target tracking, coherent motion, and obstacle evasion. Throughout their navigation, they demonstrate a nature of self-organization and self-adjustment that drives them to maintain certain social distances with each other, and adaptively adjust their behaviors in response to the environmental changes. Our findings suggest a promising approach to mitigate the spread of viral pandemics and averting stampedes.

This paper focuses on causal representation learning (CRL) under a general nonparametric causal latent model and a general transformation model that maps the latent data to the observational data. It establishes \textbf{identifiability} and \textbf{achievability} results using two hard \textbf{uncoupled} interventions per node in the latent causal graph. Notably, one does not know which pair of intervention environments have the same node intervened (hence, uncoupled environments). For identifiability, the paper establishes that perfect recovery of the latent causal model and variables is guaranteed under uncoupled interventions. For achievability, an algorithm is designed that uses observational and interventional data and recovers the latent causal model and variables with provable guarantees for the algorithm. This algorithm leverages score variations across different environments to estimate the inverse of the transformer and, subsequently, the latent variables. The analysis, additionally, recovers the existing identifiability result for two hard \textbf{coupled} interventions, that is when metadata about the pair of environments that have the same node intervened is known. It is noteworthy that the existing results on non-parametric identifiability require assumptions on interventions and additional faithfulness assumptions. This paper shows that when observational data is available, additional faithfulness assumptions are unnecessary.

Current approaches in paraphrase generation and detection heavily rely on a single general similarity score, ignoring the intricate linguistic properties of language. This paper introduces two new tasks to address this shortcoming by considering paraphrase types - specific linguistic perturbations at particular text positions. We name these tasks Paraphrase Type Generation and Paraphrase Type Detection. Our results suggest that while current techniques perform well in a binary classification scenario, i.e., paraphrased or not, the inclusion of fine-grained paraphrase types poses a significant challenge. While most approaches are good at generating and detecting general semantic similar content, they fail to understand the intrinsic linguistic variables they manipulate. Models trained in generating and identifying paraphrase types also show improvements in tasks without them. In addition, scaling these models further improves their ability to understand paraphrase types. We believe paraphrase types can unlock a new paradigm for developing paraphrase models and solving tasks in the future.

This paper proposes a new method for differentiating through optimal trajectories arising from non-convex, constrained discrete-time optimal control (COC) problems using the implicit function theorem (IFT). Previous works solve a differential Karush-Kuhn-Tucker (KKT) system for the trajectory derivative, and achieve this efficiently by solving an auxiliary Linear Quadratic Regulator (LQR) problem. In contrast, we directly evaluate the matrix equations which arise from applying variable elimination on the Lagrange multiplier terms in the (differential) KKT system. By appropriately accounting for the structure of the terms within the resulting equations, we show that the trajectory derivatives scale linearly with the number of timesteps. Furthermore, our approach allows for easy parallelization, significantly improved scalability with model size, direct computation of vector-Jacobian products and improved numerical stability compared to prior works. As an additional contribution, we unify prior works, addressing claims that computing trajectory derivatives using IFT scales quadratically with the number of timesteps. We evaluate our method on a both synthetic benchmark and four challenging, learning from demonstration benchmarks including a 6-DoF maneuvering quadrotor and 6-DoF rocket powered landing.

This paper introduces a novel adaptive transmission scheme to amplify the prowess of coordinated direct and relay transmission (CDRT) systems rooted in non-orthogonal multiple access principles. Leveraging the maximum ratio transmission scheme, we seamlessly meet the prerequisites of CDRT while harnessing the potential of dynamic power allocation and directional antennas to elevate the system's operational efficiency. Through meticulous derivations, we unveil closed-form expressions depicting the exact effective sum throughput. Our simulation results adeptly validate the theoretical analysis and vividly showcase the effectiveness of the proposed scheme.

In the classic measurement error framework, covariates are contaminated by independent additive noise. This paper considers parameter estimation in such a linear errors-in-variables model where the unknown measurement error distribution is heteroscedastic across observations. We propose a new generalized method of moment (GMM) estimator that combines a moment correction approach and a phase function-based approach. The former requires distributions to have four finite moments, while the latter relies on covariates having asymmetric distributions. The new estimator is shown to be consistent and asymptotically normal under appropriate regularity conditions. The asymptotic covariance of the estimator is derived, and the estimated standard error is computed using a fast bootstrap procedure. The GMM estimator is demonstrated to have strong finite sample performance in numerical studies, especially when the measurement errors follow non-Gaussian distributions.