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We derive optimality conditions for the optimum sample allocation problem, formulated as the determination of the fixed strata sample sizes that minimize the total cost of the survey, under assumed level of the variance of the stratified estimator and one-sided upper bounds imposed on sample sizes in strata. In this context, we take that the variance function is of some generic form that involves the stratified $\pi$ estimator of the population total with stratified simple random sampling without replacement design as a special case. The optimality conditions mentioned above will be derived with the use of convex optimization theory and the Karush-Kuhn-Tucker conditions. Based on the established optimality conditions we give a formal proof of the existing procedure, termed here as LRNA, that solves the allocation problem considered. We formulate the LRNA in such a way that it also provides the solution to classical optimum allocation problem (i.e. minimization of the estimator's variance under fixed total cost) under one-sided lower bounds imposed on sample sizes in strata. From this standpoint, the LRNA can be considered as a counterparty to the popular recursive Neyman allocation procedure that is used to solve the classical problem of optimum sample allocation but with one-sided upper bounds. Ready-to-use R-implementation of the LRNA is available through our package stratallo, which is published on the Comprehensive R Archive Network (CRAN) package repository.

Inspired by Solomonoffs theory of inductive inference, we propose a prior based on circuit complexity. There are several advantages to this approach. First, it relies on a complexity measure that does not depend on the choice of UTM. There is one universal definition for Boolean circuits involving an universal operation such as nand with simple conversions to alternative definitions such as and, or, and not. Second, there is no analogue of the halting problem. The output value of a circuit can be calculated recursively by computer in time proportional to the number of gates, while a short program may run for a very long time. Our prior assumes that a Boolean function, or equivalently, Boolean string of fixed length, is generated by some Bayesian mixture of circuits. This model is appropriate for learning Boolean functions from partial information, a problem often encountered within machine learning as "binary classification." We argue that an inductive bias towards simple explanations as measured by circuit complexity is appropriate for this problem.

This paper is concerned with a finite-horizon inverse control problem, which has the goal of inferring, from observations, the possibly non-convex and non-stationary cost driving the actions of an agent. In this context, we present a result that enables cost estimation by solving an optimization problem that is convex even when the agent cost is not and when the underlying dynamics is nonlinear, non-stationary and stochastic. To obtain this result, we also study a finite-horizon forward control problem that has randomized policies as decision variables. For this problem, we give an explicit expression for the optimal solution. Moreover, we turn our findings into algorithmic procedures and we show the effectiveness of our approach via both in-silico and experimental validations with real hardware. All the experiments confirm the effectiveness of our approach.

The stochastic block model is a canonical random graph model for clustering and community detection on network-structured data. Decades of extensive study on the problem have established many profound results, among which the phase transition at the Kesten-Stigum threshold is particularly interesting both from a mathematical and an applied standpoint. It states that no estimator based on the network topology can perform substantially better than chance on sparse graphs if the model parameter is below certain threshold. Nevertheless, if we slightly extend the horizon to the ubiquitous semi-supervised setting, such a fundamental limitation will disappear completely. We prove that with arbitrary fraction of the labels revealed, the detection problem is feasible throughout the parameter domain. Moreover, we introduce two efficient algorithms, one combinatorial and one based on optimization, to integrate label information with graph structures. Our work brings a new perspective to stochastic model of networks and semidefinite program research.

Gaussianization is a simple generative model that can be trained without backpropagation. It has shown compelling performance on low dimensional data. As the dimension increases, however, it has been observed that the convergence speed slows down. We show analytically that the number of required layers scales linearly with the dimension for Gaussian input. We argue that this is because the model is unable to capture dependencies between dimensions. Empirically, we find the same linear increase in cost for arbitrary input $p(x)$, but observe favorable scaling for some distributions. We explore potential speed-ups and formulate challenges for further research.

Transfer learning has emerged as a key approach in the machine learning domain, enabling the application of knowledge derived from one domain to improve performance on subsequent tasks. Given the often limited information about these subsequent tasks, a strong transfer learning approach calls for the model to capture a diverse range of features during the initial pretraining stage. However, recent research suggests that, without sufficient regularization, the network tends to concentrate on features that primarily reduce the pretraining loss function. This tendency can result in inadequate feature learning and impaired generalization capability for target tasks. To address this issue, we propose Variance-Covariance Regularization (VCR), a regularization technique aimed at fostering diversity in the learned network features. Drawing inspiration from recent advancements in the self-supervised learning approach, our approach promotes learned representations that exhibit high variance and minimal covariance, thus preventing the network from focusing solely on loss-reducing features. We empirically validate the efficacy of our method through comprehensive experiments coupled with in-depth analytical studies on the learned representations. In addition, we develop an efficient implementation strategy that assures minimal computational overhead associated with our method. Our results indicate that VCR is a powerful and efficient method for enhancing transfer learning performance for both supervised learning and self-supervised learning, opening new possibilities for future research in this domain.

This paper presents a novel, efficient, high-order accurate, and stable spectral element-based model for computing the complete three-dimensional linear radiation and diffraction problem for floating offshore structures. We present a solution to a pseudo-impulsive formulation in the time domain, where the frequency-dependent quantities, such as added mass, radiation damping, and wave excitation force for arbitrary heading angle, $\beta$, are evaluated using Fourier transforms from the tailored time-domain responses. The spatial domain is tessellated by an unstructured high-order hybrid configured mesh and represented by piece-wise polynomial basis functions in the spectral element space. Fourth-order accurate time integration is employed through an explicit four-stage Runge-Kutta method and complemented by fourth-order finite difference approximations for time differentiation. To reduce the computational burden, the model can make use of symmetry boundaries in the domain representation. The key piece of the numerical model -- the discrete Laplace solver -- is validated through $p$- and $h$-convergence studies. Moreover, to highlight the capabilities of the proposed model, we present prof-of-concept examples of simple floating bodies (a sphere and a box). Lastly, a much more involved case is performed of an oscillating water column, including generalized modes resembling the piston motion and wave sloshing effects inside the wave energy converter chamber. In this case, the spectral element model trivially computes the infinite-frequency added mass, which is a singular problem for conventional boundary element type solvers.

We prove that given a computable metric space and two computable measures, the set of points that have high universal uniform test scores with respect to the first measure will have a lower bound with respect to the second measure. This result is transferred to thermodynamics, showing that algorithmic thermodynamic entropy must oscillate in the presence of dynamics. Another application is that outliers will become emergent in computable dynamics of computable metric spaces.

A mainstream type of current self-supervised learning methods pursues a general-purpose representation that can be well transferred to downstream tasks, typically by optimizing on a given pretext task such as instance discrimination. In this work, we argue that existing pretext tasks inevitably introduce biases into the learned representation, which in turn leads to biased transfer performance on various downstream tasks. To cope with this issue, we propose Maximum Entropy Coding (MEC), a more principled objective that explicitly optimizes on the structure of the representation, so that the learned representation is less biased and thus generalizes better to unseen downstream tasks. Inspired by the principle of maximum entropy in information theory, we hypothesize that a generalizable representation should be the one that admits the maximum entropy among all plausible representations. To make the objective end-to-end trainable, we propose to leverage the minimal coding length in lossy data coding as a computationally tractable surrogate for the entropy, and further derive a scalable reformulation of the objective that allows fast computation. Extensive experiments demonstrate that MEC learns a more generalizable representation than previous methods based on specific pretext tasks. It achieves state-of-the-art performance consistently on various downstream tasks, including not only ImageNet linear probe, but also semi-supervised classification, object detection, instance segmentation, and object tracking. Interestingly, we show that existing batch-wise and feature-wise self-supervised objectives could be seen equivalent to low-order approximations of MEC. Code and pre-trained models are available at //github.com/xinliu20/MEC.

Graph neural networks (GNNs) are a popular class of machine learning models whose major advantage is their ability to incorporate a sparse and discrete dependency structure between data points. Unfortunately, GNNs can only be used when such a graph-structure is available. In practice, however, real-world graphs are often noisy and incomplete or might not be available at all. With this work, we propose to jointly learn the graph structure and the parameters of graph convolutional networks (GCNs) by approximately solving a bilevel program that learns a discrete probability distribution on the edges of the graph. This allows one to apply GCNs not only in scenarios where the given graph is incomplete or corrupted but also in those where a graph is not available. We conduct a series of experiments that analyze the behavior of the proposed method and demonstrate that it outperforms related methods by a significant margin.