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We rigorously study the joint evolution of training dynamics via stochastic gradient descent (SGD) and the spectra of empirical Hessian and gradient matrices. We prove that in two canonical classification tasks for multi-class high-dimensional mixtures and either 1 or 2-layer neural networks, the SGD trajectory rapidly aligns with emerging low-rank outlier eigenspaces of the Hessian and gradient matrices. Moreover, in multi-layer settings this alignment occurs per layer, with the final layer's outlier eigenspace evolving over the course of training, and exhibiting rank deficiency when the SGD converges to sub-optimal classifiers. This establishes some of the rich predictions that have arisen from extensive numerical studies in the last decade about the spectra of Hessian and information matrices over the course of training in overparametrized networks.

We present a method to improve the calibration of deep ensembles in the small training data regime in the presence of unlabeled data. Our approach is extremely simple to implement: given an unlabeled set, for each unlabeled data point, we simply fit a different randomly selected label with each ensemble member. We provide a theoretical analysis based on a PAC-Bayes bound which guarantees that if we fit such a labeling on unlabeled data, and the true labels on the training data, we obtain low negative log-likelihood and high ensemble diversity on testing samples. Empirically, through detailed experiments, we find that for low to moderately-sized training sets, our ensembles are more diverse and provide better calibration than standard ensembles, sometimes significantly.

Recently, advancements in deep learning-based superpixel segmentation methods have brought about improvements in both the efficiency and the performance of segmentation. However, a significant challenge remains in generating superpixels that strictly adhere to object boundaries while conveying rich visual significance, especially when cross-surface color correlations may interfere with objects. Drawing inspiration from neural structure and visual mechanisms, we propose a biological network architecture comprising an Enhanced Screening Module (ESM) and a novel Boundary-Aware Label (BAL) for superpixel segmentation. The ESM enhances semantic information by simulating the interactive projection mechanisms of the visual cortex. Additionally, the BAL emulates the spatial frequency characteristics of visual cortical cells to facilitate the generation of superpixels with strong boundary adherence. We demonstrate the effectiveness of our approach through evaluations on both the BSDS500 dataset and the NYUv2 dataset.

We study the complexity of the model-checking problem for discrete-timed systems with arbitrarily many anonymous and identical contributors, with and without a distinguished "controller" process, communicating via synchronous rendezvous. Our work extends the seminal work on untimed systems by German and Sistla adding discrete-time clocks, thus allowing one to model more realistic protocols. For the case without a controller, we show that the systems can be efficiently simulated -- and vice versa -- by systems of untimed processes communicating via rendezvous and symmetric broadcast, which we call "RB-systems". Symmetric broadcast is a novel communication primitive that, like ordinary asymmetric broadcast, allows all processes to synchronize without distinction between sender/receiver processes. We show that the complexity of the parameterized model-checking problem for safety specifications is pspace-complete, and for liveness specifications it is decidable in exptime. The latter result required automata theory, rational linear programming, and geometric reasoning for solving certain reachability questions in a new variant of vector addition systems called "vector rendezvous systems". We believe such proof techniques are of independent interest and will be useful in solving related problems. For the case with a controller, we show that the parameterized model-checking problems for RB-systems and systems with asymmetric broadcast are inter-reducible. This implies that for discrete timed-networks with a controller the parameterized model-checking problem is undecidable for liveness specifications. Our work exploits the intimate connection between discrete-timed systems and systems of processes communicating via broadcast. This allows us to prove decidability results for liveness properties of parameterized timed-systems, as well as extend work from untimed systems to timed systems.

Recently, Sato et al. proposed an public verifiable blind quantum computation (BQC) protocol by inserting a third-party arbiter. However, it is not true public verifiable in a sense, because the arbiter is determined in advance and participates in the whole process. In this paper, a public verifiable protocol for measurement-only BQC is proposed. The fidelity between arbitrary states and the graph states of 2-colorable graphs is estimated by measuring the entanglement witnesses of the graph states,so as to verify the correctness of the prepared graph states. Compared with the previous protocol, our protocol is public verifiable in the true sense by allowing other random clients to execute the public verification. It also has greater advantages in the efficiency, where the number of local measurements is O(n^3*log {n}) and graph states' copies is O(n^2*log{n}).

Conditional independence plays a foundational role in database theory, probability theory, information theory, and graphical models. In databases, conditional independence appears in database normalization and is known as the (embedded) multivalued dependency. Many properties of conditional independence are shared across various domains, and to some extent these commonalities can be studied through a measure-theoretic approach. The present paper proposes an alternative approach via semiring relations, defined by extending database relations with tuple annotations from some commutative semiring. Integrating various interpretations of conditional independence in this context, we investigate how the choice of the underlying semiring impacts the corresponding axiomatic and decomposition properties. We specifically identify positivity and multiplicative cancellativity as the key semiring properties that enable extending results from the relational context to the broader semiring framework. Additionally, we explore the relationships between different conditional independence notions through model theory, and consider how methods to test logical consequence and validity generalize from database theory and information theory to semiring relations.

Generalized cross-validation (GCV) is a widely-used method for estimating the squared out-of-sample prediction risk that employs a scalar degrees of freedom adjustment (in a multiplicative sense) to the squared training error. In this paper, we examine the consistency of GCV for estimating the prediction risk of arbitrary ensembles of penalized least squares estimators. We show that GCV is inconsistent for any finite ensemble of size greater than one. Towards repairing this shortcoming, we identify a correction that involves an additional scalar correction (in an additive sense) based on degrees of freedom adjusted training errors from each ensemble component. The proposed estimator (termed CGCV) maintains the computational advantages of GCV and requires neither sample splitting, model refitting, or out-of-bag risk estimation. The estimator stems from a finer inspection of ensemble risk decomposition and two intermediate risk estimators for the components in this decomposition. We provide a non-asymptotic analysis of the CGCV and the two intermediate risk estimators for ensembles of convex penalized estimators under Gaussian features and a linear response model. In the special case of ridge regression, we extend the analysis to general feature and response distributions using random matrix theory, which establishes model-free uniform consistency of CGCV.

We consider the general class of time-homogeneous stochastic dynamical systems, both discrete and continuous, and study the problem of learning a representation of the state that faithfully captures its dynamics. This is instrumental to learn the transfer operator of the system, that in turn can be used for numerous tasks, such as forecasting and interpreting the system dynamics. We show that the search for a good representation can be cast as an optimization problem over neural networks. Our approach is supported by recent results in statistical learning theory, highlighting the role of approximation error and metric distortion in the context of transfer operator regression. The objective function we propose is associated with projection operators from the representation space to the data space, overcomes metric distortion, and can be empirically estimated from data. In the discrete time setting, we further derive a relaxed objective function that is differentiable and numerically well-conditioned. We compare our method against state-of-the-art approaches on different datasets, showing better performance across the board.

We introduce a flexible method to simultaneously infer both the drift and volatility functions of a discretely observed scalar diffusion. We introduce spline bases to represent these functions and develop a Markov chain Monte Carlo algorithm to infer, a posteriori, the coefficients of these functions in the spline basis. A key innovation is that we use spline bases to model transformed versions of the drift and volatility functions rather than the functions themselves. The output of the algorithm is a posterior sample of plausible drift and volatility functions that are not constrained to any particular parametric family. The flexibility of this approach provides practitioners a powerful investigative tool, allowing them to posit a variety of parametric models to better capture the underlying dynamics of their processes of interest. We illustrate the versatility of our method by applying it to challenging datasets from finance, paleoclimatology, and astrophysics. In view of the parametric diffusion models widely employed in the literature for those examples, some of our results are surprising since they call into question some aspects of these models.

We study functional and concurrent calculi with non-determinism, along with type systems to control resources based on linearity. The interplay between non-determinism and linearity is delicate: careless handling of branches can discard resources meant to be used exactly once. Here we go beyond prior work by considering non-determinism in its standard sense: once a branch is selected, the rest are discarded. Our technical contributions are three-fold. First, we introduce a $\pi$-calculus with non-deterministic choice, governed by session types. Second, we introduce a resource $\lambda$-calculus, governed by intersection types, in which non-determinism concerns fetching of resources from bags. Finally, we connect our two typed non-deterministic calculi via a correct translation.