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The synthesis problem asks to automatically generate, if it exists, an algorithm from a specification of correct input-output pairs. In this paper, we consider the synthesis of computable functions of infinite words, for a classical Turing computability notion over infinite inputs. We consider specifications which are rational relations of infinite words, i.e., specifications defined non-deterministic parity transducers. We prove that the synthesis problem of computable functions from rational specifications is undecidable. We provide an incomplete but sound reduction to some parity game, such that if Eve wins the game, then the rational specification is realizable by a computable function. We prove that this function is even computable by a deterministic two-way transducer. We provide a sufficient condition under which the latter game reduction is complete. This entails the decidability of the synthesis problem of computable functions, which we prove to be ExpTime-complete, for a large subclass of rational specifications, namely deterministic rational specifications. This subclass contains the class of automatic relations over infinite words, a yardstick in reactive synthesis.

Discovering causal relationships from time series data is significant in fields such as finance, climate science, and neuroscience. However, contemporary techniques rely on the simplifying assumption that data originates from the same causal model, while in practice, data is heterogeneous and can stem from different causal models. In this work, we relax this assumption and perform causal discovery from time series data originating from a mixture of causal models. We propose a general variational inference-based framework called MCD to infer the underlying causal models as well as the mixing probability of each sample. Our approach employs an end-to-end training process that maximizes an evidence-lower bound for the data likelihood. We present two variants: MCD-Linear for linear relationships and independent noise, and MCD-Nonlinear for nonlinear causal relationships and history-dependent noise. We demonstrate that our method surpasses state-of-the-art benchmarks in causal discovery tasks through extensive experimentation on synthetic and real-world datasets, particularly when the data emanates from diverse underlying causal graphs. Theoretically, we prove the identifiability of such a model under some mild assumptions.

The square root velocity transformation is crucial for efficiently employing the elastic approach in functional and shape data analysis of curves. We study fundamental geometric properties of curves under this transformation. Moreover, utilizing natural geometric constructions, we employ the approach for intrinsic comparison within several classes of surfaces and augmented curves, which arise in the real world applications such as tubes, ruled surfaces spherical strips, protein molecules and hurricane tracks.

This study investigates the asymptotic dynamics of alternating minimization applied to optimize a bilinear non-convex function with normally distributed covariates. We employ the replica method from statistical physics in a multi-step approach to precisely trace the algorithm's evolution. Our findings indicate that the dynamics can be described effectively by a two--dimensional discrete stochastic process, where each step depends on all previous time steps, revealing a memory dependency in the procedure. The theoretical framework developed in this work is broadly applicable for the analysis of various iterative algorithms, extending beyond the scope of alternating minimization.

Most mathematical distortions used in ML are fundamentally integral in nature: $f$-divergences, Bregman divergences, (regularized) optimal transport distances, integral probability metrics, geodesic distances, etc. In this paper, we unveil a grounded theory and tools which can help improve these distortions to better cope with ML requirements. We start with a generalization of Riemann integration that also encapsulates functions that are not strictly additive but are, more generally, $t$-additive, as in nonextensive statistical mechanics. Notably, this recovers Volterra's product integral as a special case. We then generalize the Fundamental Theorem of calculus using an extension of the (Euclidean) derivative. This, along with a series of more specific Theorems, serves as a basis for results showing how one can specifically design, alter, or change fundamental properties of distortion measures in a simple way, with a special emphasis on geometric- and ML-related properties that are the metricity, hyperbolicity, and encoding. We show how to apply it to a problem that has recently gained traction in ML: hyperbolic embeddings with a "cheap" and accurate encoding along the hyperbolic vs Euclidean scale. We unveil a new application for which the Poincar\'e disk model has very appealing features, and our theory comes in handy: \textit{model} embeddings for boosted combinations of decision trees, trained using the log-loss (trees) and logistic loss (combinations).

We conduct a systematic study of the approximation properties of Transformer for sequence modeling with long, sparse and complicated memory. We investigate the mechanisms through which different components of Transformer, such as the dot-product self-attention, positional encoding and feed-forward layer, affect its expressive power, and we study their combined effects through establishing explicit approximation rates. Our study reveals the roles of critical parameters in the Transformer, such as the number of layers and the number of attention heads, and these insights also provide natural suggestions for alternative architectures.

We present a theoretical approach to overcome the curse of dimensionality using a neural computation algorithm which can be distributed across several machines. Our modular distributed deep learning paradigm, termed \textit{neural pathways}, can achieve arbitrary accuracy while only loading a small number of parameters into GPU VRAM. Formally, we prove that for every error level $\varepsilon>0$ and every Lipschitz function $f:[0,1]^n\to \mathbb{R}$, one can construct a neural pathways model which uniformly approximates $f$ to $\varepsilon$ accuracy over $[0,1]^n$ while only requiring networks of $\mathcal{O}(\varepsilon^{-1})$ parameters to be loaded in memory and $\mathcal{O}(\varepsilon^{-1}\log(\varepsilon^{-1}))$ to be loaded during the forward pass. This improves the optimal bounds for traditional non-distributed deep learning models, namely ReLU MLPs, which need $\mathcal{O}(\varepsilon^{-n/2})$ parameters to achieve the same accuracy. The only other available deep learning model that breaks the curse of dimensionality is MLPs with super-expressive activation functions. However, we demonstrate that these models have an infinite VC dimension, even with bounded depth and width restrictions, unlike the neural pathways model. This implies that only the latter generalizes. Our analysis is validated experimentally in both regression and classification tasks, demonstrating that our model exhibits superior performance compared to larger centralized benchmarks.

We consider approximating so-called tame functions, a class of nonsmooth, nonconvex functions, with piecewise polynomial functions. Tame functions appear in a wide range of applications: functions encountered in the training of deep neural networks with all common activations, value functions of mixed-integer programs, or wave functions of small molecules. We bound the quality of approximation of a tame function by a piecewise polynomial function with a given number of segments on any full-dimensional cube. We also present the first ever mixed-integer programming formulation of piecewise polynomial regression. Together, these can be used to estimate tame functions. We demonstrate promising computational results.

We describe a version of the FGLM algorithm that can be used to compute generic fibers of positive-dimensional polynomial ideals. It combines the FGLM algorithm with a Hensel lifting strategy. We show that this algorithm has a complexity quasi-linear in the number of lifting steps. Some provided experimental data also demonstrates the practical efficacy of our algorithm. Additionally, we sketch a related Hensel lifting method to compute Gr\"obner bases using so-called tracers.

Evolutionary Computation algorithms have been used to solve optimization problems in relation with architectural, hyper-parameter or training configuration, forging the field known today as Neural Architecture Search. These algorithms have been combined with other techniques such as the pruning of Neural Networks, which reduces the complexity of the network, and the Transfer Learning, which lets the import of knowledge from another problem related to the one at hand. The usage of several criteria to evaluate the quality of the evolutionary proposals is also a common case, in which the performance and complexity of the network are the most used criteria. This work proposes MO-EvoPruneDeepTL, a multi-objective evolutionary pruning algorithm. MO-EvoPruneDeepTL uses Transfer Learning to adapt the last layers of Deep Neural Networks, by replacing them with sparse layers evolved by a genetic algorithm, which guides the evolution based in the performance, complexity and robustness of the network, being the robustness a great quality indicator for the evolved models. We carry out different experiments with several datasets to assess the benefits of our proposal. Results show that our proposal achieves promising results in all the objectives, and direct relation are presented among them. The experiments also show that the most influential neurons help us explain which parts of the input images are the most relevant for the prediction of the pruned neural network. Lastly, by virtue of the diversity within the Pareto front of pruning patterns produced by the proposal, it is shown that an ensemble of differently pruned models improves the overall performance and robustness of the trained networks.