Probabilistic numerics casts numerical tasks, such the numerical solution of differential equations, as inference problems to be solved. One approach is to model the unknown quantity of interest as a random variable, and to constrain this variable using data generated during the course of a traditional numerical method. However, data may be nonlinearly related to the quantity of interest, rendering the proper conditioning of random variables difficult and limiting the range of numerical tasks that can be addressed. Instead, this paper proposes to construct probabilistic numerical methods based only on the final output from a traditional method. A convergent sequence of approximations to the quantity of interest constitute a dataset, from which the limiting quantity of interest can be extrapolated, in a probabilistic analogue of Richardson's deferred approach to the limit. This black box approach (1) massively expands the range of tasks to which probabilistic numerics can be applied, (2) inherits the features and performance of state-of-the-art numerical methods, and (3) enables provably higher orders of convergence to be achieved. Applications are presented for nonlinear ordinary and partial differential equations, as well as for eigenvalue problems-a setting for which no probabilistic numerical methods have yet been developed.
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Recent advances in deep generative models have led to impressive results in a variety of application domains. Motivated by the possibility that deep learning models might memorize part of the input data, there have been increased efforts to understand how memorization arises. In this work, we extend a recently proposed measure of memorization for supervised learning (Feldman, 2019) to the unsupervised density estimation problem and adapt it to be more computationally efficient. Next, we present a study that demonstrates how memorization can occur in probabilistic deep generative models such as variational autoencoders. This reveals that the form of memorization to which these models are susceptible differs fundamentally from mode collapse and overfitting. Furthermore, we show that the proposed memorization score measures a phenomenon that is not captured by commonly-used nearest neighbor tests. Finally, we discuss several strategies that can be used to limit memorization in practice. Our work thus provides a framework for understanding problematic memorization in probabilistic generative models.
We reexamine the the classical multidimensional scaling (MDS). We study some special cases, in particular, the exact solution for the sub-space formed by the 3 dimensional principal coordinates is derived. Also we give the extreme case when the points are collinear. Some insight into the effect on the MDS solution of the excluded eigenvalues (could be both positive as well as negative) of the doubly centered matrix is provided. As an illustration, we work through an example to understand the distortion in the MDS construction with positive and negative eigenvalues.
In this paper, we study deep neural networks (DNNs) for solving high-dimensional evolution equations with oscillatory solutions. Different from deep least-squares methods that deal with time and space variables simultaneously, we propose a deep adaptive basis Galerkin (DABG) method which employs the spectral-Galerkin method for time variable by tensor-product basis for oscillatory solutions and the deep neural network method for high-dimensional space variables. The proposed method can lead to a linear system of differential equations having unknown DNNs that can be trained via the loss function. We establish a posterior estimates of the solution error which is bounded by the minimal loss function and the term $O(N^{-m})$, where $N$ is the number of basis functions and $m$ characterizes the regularity of the equation, and show that if the true solution is a Barron-type function, the error bound converges to zero as $M=O(N^p)$ approaches to infinity where $M$ is the width of the used networks and $p$ is a positive constant. Numerical examples including high-dimensional linear parabolic and hyperbolic equations, and nonlinear Allen-Cahn equation are presented to demonstrate the performance of the proposed DABG method is better than that of existing DNNs.
This paper deals with a special type of Lyapunov functions, namely the solution of Zubov's equation. Such a function can be used to characterize the domain of attraction for systems of ordinary differential equations. We derive and prove an integral form solution to Zubov's equation. For numerical computation, we develop two data-driven methods. One is based on the integration of an augmented system of differential equations; and the other one is based on deep learning. The former is effective for systems with a relatively low state space dimension and the latter is developed for high dimensional problems. The deep learning method is applied to a New England 10-generator power system model. We prove that a neural network approximation exists for the Lyapunov function of power systems such that the approximation error is a cubic polynomial of the number of generators. The error convergence rate as a function of n, the number of neurons, is proved.
This study concerns probability distribution estimation of sample maximum. The traditional approach is the parametric fitting to the limiting distribution - the generalized extreme value distribution; however, the model in finite cases is misspecified to a certain extent. We propose a plug-in type of the kernel distribution estimator which does not need model specification. It is proved that both asymptotic convergence rates depend on the tail index and the second order parameter. As the tail gets light, the degree of misspecification of the parametric fitting becomes large, that means the convergence rate becomes slow. In the Weibull cases, which can be seen as the limit of tail-lightness, only the nonparametric distribution estimator keeps its consistency. Finally, we report results of numerical experiments and two real case studies.
Let $Q_{n}^{r}$ be the graph with vertex set $\{-1,1\}^{n}$ in which two vertices are joined if their Hamming distance is at most $r$. The edge-isoperimetric problem for $Q_{n}^{r}$ is that: For every $(n,r,M)$ such that $1\le r\le n$ and $1\le M\le2^{n}$, determine the minimum edge-boundary size of a subset of vertices of $Q_{n}^{r}$ with a given size $M$. In this paper, we apply two different approaches to prove bounds for this problem. The first approach is a linear programming approach and the second is a probabilistic approach. Our bound derived by the first approach generalizes the tight bound for $M=2^{n-1}$ derived by Kahn, Kalai, and Linial in 1989. Moreover, our bound is also tight for $M=2^{n-2}$ and $r\le\frac{n}{2}-1$. Our bounds derived by the second approach are expressed in terms of the \emph{noise stability}, and they are shown to be asymptotically tight as $n\to\infty$ when $r=2\lfloor\frac{\beta n}{2}\rfloor+1$ and $M=\lfloor\alpha2^{n}\rfloor$ for fixed $\alpha,\beta\in(0,1)$, and is tight up to a factor $2$ when $r=2\lfloor\frac{\beta n}{2}\rfloor$ and $M=\lfloor\alpha2^{n}\rfloor$. In fact, the edge-isoperimetric problem is equivalent to a ball-noise stability problem which is a variant of the traditional (i.i.d.-) noise stability problem. Our results can be interpreted as bounds for the ball-noise stability problem.
The virtual element method (VEM) is a Galerkin approximation method that extends the finite element method to polytopal meshes. In this paper, we present two different conforming virtual element formulations for the numerical approximation of the Stokes problem that work on polygonal meshes.The velocity vector field is approximated in the virtual element spaces of the two formulations, while the pressure variable is approximated through discontinuous polynomials. Both formulations are inf-sup stable and convergent with optimal convergence rates in the $L^2$ and energy norm. We assess the effectiveness of these numerical approximations by investigating their behavior on a representative benchmark problem. The observed convergence rates are in accordance with the theoretical expectations and a weak form of the zero-divergence constraint is satisfied at the machine precision level.
In order to gain a better understanding of the state space of programs, with the aim of making their verification more tractable, models based on directed topological spaces have been introduced, allowing to take in account equivalence between execution traces, as well as translate features of the execution (such as the presence of deadlocks) into geometrical situations. In this context, many algorithms were introduced, based on a description of the geometrical models as regions consisting of unions of rectangles. We explain here that these constructions can actually be performed directly on the syntax of programs, thus resulting in representations which are more natural and easier to implement. In order to do so, we start from the observation that positions in a program can be described as partial explorations of the program. The operational semantics induces a partial order on positions, and regions can be defined as formal unions of intervals in the resulting poset. We then study the structure of such regions and show that, under reasonable conditions, they form a boolean algebra and admit a representation in normal form (which corresponds to covering a space by maximal intervals), thus supporting the constructions needed for the purpose of studying programs. All the operations involved here are given explicit algorithmic descriptions.
Knowledge graph reasoning, which aims at predicting the missing facts through reasoning with the observed facts, is critical to many applications. Such a problem has been widely explored by traditional logic rule-based approaches and recent knowledge graph embedding methods. A principled logic rule-based approach is the Markov Logic Network (MLN), which is able to leverage domain knowledge with first-order logic and meanwhile handle their uncertainty. However, the inference of MLNs is usually very difficult due to the complicated graph structures. Different from MLNs, knowledge graph embedding methods (e.g. TransE, DistMult) learn effective entity and relation embeddings for reasoning, which are much more effective and efficient. However, they are unable to leverage domain knowledge. In this paper, we propose the probabilistic Logic Neural Network (pLogicNet), which combines the advantages of both methods. A pLogicNet defines the joint distribution of all possible triplets by using a Markov logic network with first-order logic, which can be efficiently optimized with the variational EM algorithm. In the E-step, a knowledge graph embedding model is used for inferring the missing triplets, while in the M-step, the weights of logic rules are updated based on both the observed and predicted triplets. Experiments on multiple knowledge graphs prove the effectiveness of pLogicNet over many competitive baselines.
We present an implementation of a probabilistic first-order logic called TensorLog, in which classes of logical queries are compiled into differentiable functions in a neural-network infrastructure such as Tensorflow or Theano. This leads to a close integration of probabilistic logical reasoning with deep-learning infrastructure: in particular, it enables high-performance deep learning frameworks to be used for tuning the parameters of a probabilistic logic. Experimental results show that TensorLog scales to problems involving hundreds of thousands of knowledge-base triples and tens of thousands of examples.
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