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Spatial Monte Carlo integration (SMCI) is an extension of standard Monte Carlo integration and can approximate expectations on Markov random fields with high accuracy. SMCI was applied to pairwise Boltzmann machine (PBM) learning, achieving superior results over those of some existing methods. The approximation level of SMCI can be altered, and it was proved that a higher-order approximation of SMCI is statistically more accurate than a lower-order approximation. However, SMCI as proposed in previous studies suffers from a limitation that prevents the application of a higher-order method to dense systems. This study makes two contributions. First, a generalization of SMCI (called generalized SMCI (GSMCI)) is proposed, which allows a relaxation of the above-mentioned limitation; moreover, a statistical accuracy bound of GSMCI is proved. Second, a new PBM learning method based on SMCI is proposed, which is obtained by combining SMCI and persistent contrastive divergence. The proposed learning method significantly improves learning accuracy.A new network with super-approximation power is introduced. This network is built with Floor ( ⌊ x ⌋ ) or ReLU ( max 0 , x ) activation function in each neuron; hence, we call such networks Floor-ReLU networks. For any hyperparameters N ∈ N + and L ∈ N + , we show that Floor-ReLU networks with width max d , 5 N + 13 and depth 64 d L + 3 can uniformly approximate a Hölder function f on [ 0 , 1 ] d with an approximation error 3 λ d α / 2 N - α L , where α ∈ ( 0 , 1 ] and λ are the Hölder order and constant, respectively. More generally for an arbitrary continuous function f on [ 0 , 1 ] d with a modulus of continuity ω f ( · ) , the constructive approximation rate is ω f ( d N - L ) + 2 ω f ( d ) N - L . As a consequence, this new class of networks overcomes the curse of dimensionality in approximation power when the variation of ω f ( r ) as r → 0 is moderate (e.g., ω f ( r ) ≲ r α for Hölder continuous functions), since the major term to be considered in our approximation rate is essentially d times a function of N and L independent of d within the modulus of continuity.Sustained attention is a cognitive ability to maintain task focus over extended periods of time (Mackworth, 1948; Chun, Golomb, & Turk-Browne, 2011). Monocrotaline supplier In this study, scalp electroencephalography (EEG) signals were processed in real time using a 32 dry-electrode system during a sustained visual attention task. An attention training paradigm was implemented, as designed in DeBettencourt, Cohen, Lee, Norman, and Turk-Browne (2015) in which the composition of a sequence of blended images is updated based on the participant's decoded attentional level to a primed image category. It was hypothesized that a single neurofeedback training session would improve sustained attention abilities. Twenty-two participants were trained on a single neurofeedback session with behavioral pretraining and posttraining sessions within three consecutive days. Half of the participants functioned as controls in a double-blinded design and received sham neurofeedback. During the neurofeedback session, attentional states to primed categorPython based and open source, and it allows users to actively engage in the development of neurofeedback tools for scientific and translational use.In this note, I study how the precision of a binary classifier depends on the ratio r of positive to negative cases in the test set, as well as the classifier's true and false-positive rates. This relationship allows prediction of how the precision-recall curve will change with r , which seems not to be well known. It also allows prediction of how F β and the precision gain and recall gain measures of Flach and Kull (2015) vary with r .Empirical estimates of the dimensionality of neural population activity are often much lower than the population size. Similar phenomena are also observed in trained and designed neural network models. These experimental and computational results suggest that mapping low-dimensional dynamics to high-dimensional neural space is a common feature of cortical computation. Despite the ubiquity of this observation, the constraints arising from such mapping are poorly understood. Here we consider a specific example of mapping low-dimensional dynamics to high-dimensional neural activity-the neural engineering framework. We analytically solve the framework for the classic ring model-a neural network encoding a static or dynamic angular variable. Our results provide a complete characterization of the success and failure modes for this model. Based on similarities between this and other frameworks, we speculate that these results could apply to more general scenarios.Stable concurrent learning and control of dynamical systems is the subject of adaptive control. Despite being an established field with many practical applications and a rich theory, much of the development in adaptive control for nonlinear systems revolves around a few key algorithms. By exploiting strong connections between classical adaptive nonlinear control techniques and recent progress in optimization and machine learning, we show that there exists considerable untapped potential in algorithm development for both adaptive nonlinear control and adaptive dynamics prediction. We begin by introducing first-order adaptation laws inspired by natural gradient descent and mirror descent. We prove that when there are multiple dynamics consistent with the data, these non-Euclidean adaptation laws implicitly regularize the learned model. Local geometry imposed during learning thus may be used to select parameter vectors-out of the many that will achieve perfect tracking or prediction-for desired properties such as sparsity. We apply this result to regularized dynamics predictor and observer design, and as concrete examples, we consider Hamiltonian systems, Lagrangian systems, and recurrent neural networks. We subsequently develop a variational formalism based on the Bregman Lagrangian. We show that its Euler Lagrange equations lead to natural gradient and mirror descent-like adaptation laws with momentum, and we recover their first-order analogues in the infinite friction limit. We illustrate our analyses with simulations demonstrating our theoretical results.
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