This paper aims to understand the training solution, which is obtained by the back-propagation algorithm, of two-layer neural networks whose hidden layer is composed of the units with smooth activation functions, including the usual sigmoid type most commonly used before the advent of ReLUs. The mechanism contains four main principles: construction of Taylor series expansions, strict partial order of knots, smooth-spline implementation and smooth-continuity restriction. The universal approximation for arbitrary input dimensionality is proved and experimental verification is given, through which the mystery of ``black box'' of the solution space is largely revealed. The new proofs employed also enrich approximation theory.

A neural network with one hidden layer or a two-layer network (regardless of the input layer) is the simplest feedforward neural network, whose mechanism may be the basis of more general network architectures. However, even to this type of simple architecture, it is also a ``black box''; that is, it remains unclear how to interpret the mechanism of its solutions obtained by the back-propagation algorithm and how to control the training process through a deterministic way. This paper systematically studies the first problem by constructing universal function-approximation solutions. It is shown that, both theoretically and experimentally, the training solution for the one-dimensional input could be completely understood, and that for a higher-dimensional input can also be well interpreted to some extent. Those results pave the way for thoroughly revealing the black box of two-layer ReLU networks and advance the understanding of deep ReLU networks.

In this paper, we take first steps toward developing defeasible reasoning on concepts in KLM framework. We define generalizations of cumulative reasoning system C and cumulative reasoning system with loop CL to conceptual setting. We also generalize cumulative models, cumulative ordered models, and preferential models to conceptual setting and show the soundness and completeness results for these models.

We present a method for hierarchical clustering of directed acyclic graphs and other strictly partially ordered data that preserves the data structure. In particular, if we have $a

Methods for choosing from a set of options are often based on a strict partial order on these options, or on a set of such partial orders. I here provide a very general axiomatic characterisation for choice functions of this form. It includes as special cases axiomatic characterisations for choice functions based on (sets of) total orders, (sets of) weak orders, (sets of) coherent lower previsions and (sets of) probability measures.

Strict partial order is a mathematical structure commonly seen in relational data. One obstacle to extracting such type of relations at scale is the lack of large-scale labels for building effective data-driven solutions. We develop an active learning framework for mining such relations subject to a strict order. Our approach incorporates relational reasoning not only in finding new unlabeled pairs whose labels can be deduced from an existing label set, but also in devising new query strategies that consider the relational structure of labels. Our experiments on concept prerequisite relations show our proposed framework can substantially improve the classification performance with the same query budget compared to other baseline approaches.

Starting with a likelihood or preference order on worlds, we extend it to a likelihood ordering on sets of worlds in a natural way, and examine the resulting logic. Lewis (1973) earlier considered such a notion of relative likelihood in the context of studying counterfactuals, but he assumed a total preference order on worlds. Complications arise when examining partial orders that are not present for total orders. There are subtleties involving the exact approach to lifting the order on worlds to an order on sets of worlds. In addition, the axiomatization of the logic of relative likelihood in the case of partial orders gives insight into the connection between relative likelihood and default reasoning.

In a standard possibilistic logic, prioritized information are encoded by means of weighted knowledge base. This paper proposes an extension of possibilistic logic for dealing with partially ordered information. We Show that all basic notions of standard possibilitic logic (sumbsumption, syntactic and semantic inference, etc.) have natural couterparts when dealing with partially ordered information. We also propose an algorithm which computes possibilistic conclusions of a partial knowledge base of a partially ordered knowlege base.

We study a dominance relation for comparing outcomes based on unconditional qualitative preferences and compare it with its unconditional counterparts for TCP-nets and their variants. Dominance testing based on this relation can be carried out in polynomial time by evaluating the satisfiability of a logic formula.

Preference queries are relational algebra or SQL queries that contain occurrences of the winnow operator ("find the most preferred tuples in a given relation"). Such queries are parameterized by specific preference relations. Semantic optimization techniques make use of integrity constraints holding in the database. In the context of semantic optimization of preference queries, we identify two fundamental properties: containment of preference relations relative to integrity constraints and satisfaction of order axioms relative to integrity constraints. We show numerous applications of those notions to preference query evaluation and optimization. As integrity constraints, we consider constraint-generating dependencies, a class generalizing functional dependencies. We demonstrate that the problems of containment and satisfaction of order axioms can be captured as specific instances of constraint-generating dependency entailment. This makes it possible to formulate necessary and sufficient conditions for the applicability of our techniques as constraint validity problems. We characterize the computational complexity of such problems.