This paper proposes KAN/H, a variant of Kolmogorov-Arnold Network (KAN) that uses a Haar-variant basis system having both global and local bases instead of B-spline. The resulting algorithm is applied to function approximation problems and MNIST. We show that it does not require most of the problem-specific hyper-parameter tunings.

Theorem 1. Suppose that ˆ X In DB models, the commonly used p is either 1 or 2. When p = 2, DURA takes the form as the one in Equation (8) in the main text. If p = 1, we cannot expand the squared score function of the associated DB models as in Equation (4). Therefore, we choose p = 2 . 2 Table 2: Hyperparameters found by grid search. Suppose that k is the number of triplets known to be true in the knowledge graph, n is the embedding dimension of entities. That is to say, the computational complexity of weighted DURA is the same as the weighted squared Frobenius norm regularizer.

Table 4: The detailed sampling hyperparameters for Chameleon.

Splitting a logic program allows us to reduce the task of computing its stable models to similar tasks for its subprograms. This can be used to increase solving performance and prove program correctness. We generalize the conditions under which this technique is applicable, by considering not only dependencies between predicates but also their arguments and context. This allows splitting programs commonly used in practice to which previous results were not applicable.

We prove the first generalization bound for large-margin halfspaces that is asymptotically tight in the tradeoff between the margin, the fraction of training points with the given margin, the failure probability and the number of training points.

In temporal extensions of Answer Set Programming (ASP) based on linear-time, the behavior of dynamic systems is captured by sequences of states. While this representation reflects their relative order, it abstracts away the specific times associated with each state. However, timing constraints are important in many applications like, for instance, when planning and scheduling go hand in hand. We address this by developing a metric extension of linear-time temporal equilibrium logic, in which temporal operators are constrained by intervals over natural numbers. The resulting Metric Equilibrium Logic provides the foundation of an ASP-based approach for specifying qualitative and quantitative dynamic constraints. To this end, we define a translation of metric formulas into monadic first-order formulas and give a correspondence between their models in Metric Equilibrium Logic and Monadic Quantified Equilibrium Logic, respectively. Interestingly, our translation provides a blue print for implementation in terms of ASP modulo difference constraints.

This paper studies fixed step-size stochastic approximation (SA) schemes, including stochastic gradient schemes, in a Riemannian framework. It is motivated by several applications, where geodesics can be computed explicitly, and their use accelerates crude Euclidean methods. A fixed step-size scheme defines a family of time-homogeneous Markov chains, parametrized by the step-size. Here, using this formulation, non-asymptotic performance bounds are derived, under Lyapunov conditions. Then, for any step-size, the corresponding Markov chain is proved to admit a unique stationary distribution, and to be geometrically ergodic. This result gives rise to a family of stationary distributions indexed by the step-size, which is further shown to converge to a Dirac measure, concentrated at the solution of the problem at hand, as the step-size goes to 0. Finally, the asymptotic rate of this convergence is established, through an asymptotic expansion of the bias, and a central limit theorem.