This paper examines the computational and sample complexity of answering k-wise statistical queries, which were introduced by Felman and Ghazi [9] as a generalization to the standard statistical query model of Kearns [11]. In particular, our paper studies two sample reuse schemes: (1) reusing independent “pseudo-samples” for adaptive queries and (2) reusing dependent k-wise samples for non-adaptive queries. Comparing to a baseline non-reuse strategy, we show that the first reuse method offers a trade-off between k, the arity of the query, and M, the total number of queries to be answered. We also show that the second reuse method performs no worse than the baseline, and possibly better, from the perspective of variance reduction.
In this paper, we study a variety of NP-hard optimization problems, such as MAXCUT, MAXkSAT, and MAXNAE2SAT from the perspective of obtaining exact solutions. We derive differentiable functions for each of these problems using the dataless neural networks framework. Recently, it was shown that a single differentiable function on a dataless neural network can capture the Maximum Independent Set problem. Inspired by this design, we design dataless neural networks for a host of combinatorial optimization problems. We also establish the correctness of our derivations in a rigorous fashion.
Long-distance resolution for quantified boolean formula (QBF) solving was introduced in 2002 by Zhang and Malik, but has been controversial for the following decade, because it derives and uses tautologous clauses. Balabanov and Jiang (2012) gave a set of proof rules (called LDQ-resolution) that include long-distance resolution with conditions, but did not attach any “meaning” (i.e., semantic interpretation) to the derived tautologous clauses. Egly, Lonsing and Widl (2013) showed that the QBF certificate extraction algorithm of Goutltiaeva, Van Gelder and Bacchus (2011) could be applied with correct results to LDQ refutations. These results and others have brought LDQ-resolution back into the mainstream. This paper introduces partial boolean functions (pbfs) and develops a semantic interpretation for universal literals in QBF clauses. It develops LDP-resolution, which is refutationally complete for QBF formulas in prenex conjunction normal form (PCNF). LDP-resolution is shown to derive only logical consequences.
In computed tomography, the reconstructed object and the measured detector values are represented in a discrete domain. As the real world is not discrete, this restriction of the model usually creates artifacts (i.e., interpolation errors) on the reconstructed images. To formulate the projection model, we use so-called interpolation methods which enumerate the interactions between beams and the image as rays pass through the pixels of the image. The type of interpolation method we select strongly influences the artifacts in the reconstructed image. In this paper, we show a connection between pixel coverage to get a better understanding of interpolation errors, and we also propose an effective correction method to reduce the effects of interpolation errors in reconstructed images. We tested our proposed method in a comprehensive experiment, where we found that our proposed correction method can significantly improve the quality of the image.
In this paper, we establish the computational complexities of selected forms of refutations of linear programs. Linear programming is in the complexity class P and hence, it must have short affirmative and disqualifying certificates. One of the more celebrated lemmata in linear programming is Farkas’ lemma, which establishes that both “yes" and “no" certificates can be thought of as solutions to complementary linear programs. Since then, it has been established that if a linear program is feasible, then it must have a solution which is bounded by a polynomial function of the input size. The latter observation, coupled with Farkas’ lemma, immediately establishes that linear programming is in NP∩coNP. Our goal is to study the computational complexities of determining various constrained refutations for a given linear programming problem. This paper focuses on three distinct refutation forms, viz., read-once, tree-like and dag-like. We establish that checking if a linear program has a read-once refutation is NP-complete, even when it is defined by Binary Two Variable Per Inequality (BTVPI) constraints. Furthermore, the problems of finding the shortest tree-like and dag-like refutations are NPO-complete and NPO PB-complete respectively.
Large-scale deployed learning systems are often evaluated along multiple objectives or criteria. But, how can we learn or optimize such complex systems, with potentially conflicting or even incompatible objectives? How can we improve the system when user feedback becomes available, feedback possibly alerting to issues not previously optimized for by the system? We present a new theoretical model for learning and optimizing such complex systems. Rather than committing to a static or pre-defined tradeoff for the multiple objectives, our model is guided by the feedback received, which is used to update its internal state. Our model supports multiple objectives that can be of very general form and takes into account their potential incompatibilities. We consider both a stochastic and an adversarial setting. In the stochastic setting, we show that our framework can be naturally cast as a Markov Decision Process with stochastic losses, for which we give efficient vanishing regret algorithmic solutions. In the adversarial setting, we design efficient algorithms with competitive ratio guarantees. We also report the results of experiments with our stochastic algorithms validating their effectiveness.
A combinatorial algorithm is presented here to determine the straight skeleton of a given 3D orthogonal polyhedron in O(nlogn) time, where n is the number of vertices in the 3D orthogonal polyhedron. The 3D orthogonal polyhedron is traversed by considering a cuboid at each step. One of the combinatorial rule is applied and a part of the 3D orthogonal polyhedron is discarded as per the reduction rule. When the whole 3D polyhedron is traversed, the straight skeleton is obtained. The straight skeleton has many applications in shape analysis, shape matching, and shape retrieval.
Constructive definitions, including inductive and recursive definitions, are ubiquitous in mathematical texts and occur in a wide variety of computer science fields and Knowledge Representation applications. While in different areas there is a high level of familiarity with certain types of constructive definitions, fairly little interaction between different areas seems to exist, resulting in a lack of deep understanding of principles and their applications. This paper aims to fill this void by laying the foundations for a single unifying framework, bringing together a wide variety of definitions. First, we recall the principle of (monotone) inductive definition and its formalization in fixpoint theory. We discuss the constructive and the non-constructive interpretation of inductive definitions and the induction process. We then analyze examples, including but not limited to (co)inductive and (co)recursive definitions, found in a wide range of areas through the lens of our proposed framework.
In this paper we analyze online problems from the perspective of when to switch solutions when the cost is of doing so is high – we call such a solution change a “recalculation.” We analyze this problem under the assumption we have algorithms that achieve per-round regret (which can be otherwise thought of as point-wise error, or other well-studied quantities) of the form O(1/tε) after seeing t data-points. We study schedules with a constant number and an increasing number of recalculations in the total number of datapoints, and we examine when achieving optimal cumulative regret is possible.
The study of the Clique problem in algorithmic graph theory is important both because it is a central problem in complexity theory, almost at the same level as SAT [8], but also because its practical resolution has many applications, notably in Artificial Intelligence (e.g. checking consistency of a binary CSP is equivalent to check the size of maximum clique in its microstructure [7]). A great deal of work has therefore been carried out, and this paper attempts to extend the results obtained in this field. It starts from the observation that this problem is tractable in polynomial time on graphs whose treewidth [10] are bounded by a constant (see [5]). Although this type of result is very interesting from a theoretical point of view, it often remains limited in terms of application. So, we propose here an extension of such these approaches based on the definition of graph classes wider than those of bounded treewidth for which we would be able to propose polynomial time algorithms. These graph classes denoted CWk include graphs of treewidth W+k, but also contain, even if k and W are constants, graphs known to be of unbounded treewidth. These CWk classes are introduced, their fundamental properties are given and the associated algorithms are presented and analysed.
The quality of components produced through additive manufacturing processes, such as selective laser melting (SLM), is significantly influenced by heat transfer phenomena. Numerical simulations have emerged as valuable tools for gaining a deeper understanding of these processes. Deep investigation is made possible by a large amount of sensor data in this area. Both offers the potential to reduce the cost and time associated with empirical experimentation. Physics-informed neural networks (PINNs) combine the data-driven capabilities of deep neural networks with the mathematical formulations of physical laws, such as heat diffusion. In particular, the gap between numerical simulations and data observations can be bridged. In this paper, we present a novel neural diffusion graph convolutional network (NDGCN) designed to reveal physically interpretable parameters and accurately predict heat transfer dynamics during the SLM process. Our methodology involves representing the fabricated part as a graph model, constructed from high-dimensional data. This facilitates the integration of complex geometries and thermal properties into our predictive framework.
We present a study of surrogate losses and algorithms for the general problem of learning to defer with multiple experts. We first introduce a new family of surrogate losses specifically tailored for the multiple-expert setting, where the prediction and deferral functions are learned simultaneously. We then prove that these surrogate losses benefit from strong H-consistency bounds. We illustrate the application of our analysis through several examples of practical surrogate losses, for which we give explicit guarantees. These loss functions readily lead to the design of new learning to defer algorithms based on their minimization. While the main focus of this work is a theoretical analysis, we also report the results of several experiments on SVHN and CIFAR-10 datasets.
Discrimination-aware data mining is expected to play an important role in data-driven decision making, as “BIG data” can be obtained from the actual society. To build the appropriate decision making system, AI researchers and practitioners have proposed various discrimination measures. However, most of the existing discrimination measures cannot be interpreted as the “proportion” and thus may not provide the comparable evaluation of the discrimination level. To evaluate how much of discrimination is based on a sensitive feature directly, indirectly, or totally, we propose three proportion measures of discrimination using natural direct and indirect effects [12]. The effectiveness of the proposed discrimination measures is confirmed on Adult Census Data [2].
In this paper we present a player for incomplete information card games with tricks scored by points or eyes. We factorize existing algorithms into a template that captures the main ingredients of such game trees. We then analyze three different algorithms, and the impact of the information given during the algorithm on the decisions it makes. We extend these algorithms to work with cost instead of winning vectors, and illustrate their effectiveness in finding good cards. We develop a new algorithm that tries to respect known information.
Too few general constructions of Boolean functions satisfying all cryptographic criteria are known. We investigate the construction in which the support of f equals the image set of an injective vectorial function F, that we call a parameterization of f. Every balanced Boolean function can be obtained this way. We study five illustrations of this general construction. The three first correspond to known classes (Maiorana-McFarland, majority functions and balanced functions in odd numbers of variables with optimal algebraic immunity). The two last correspond to new classes: - the sums of indicators of disjoint graphs of (k, n - k) -functions, - functions parameterized through (n - 1, n) -functions due to Beelen and Leander. We study the cryptographic parameters of balanced Boolean functions, according to those of their parameterizations: the algebraic degree of f, that we relate to the algebraic degrees of F and of its graph indicator, the nonlinearity of f, that we relate by a bound to the nonlinearity of F, and the algebraic immunity (AI), whose optimality is related to a natural question in linear algebra, and which may be approached (in two ways) by using the graph indicator of F. We revisit each of the five classes for each criterion. The fourth class is very promising, thanks to a lower bound on the nonlinearity by means of the nonlinearity of the chosen (k, n-k )-functions. The sub -class of the sums of indicators of affine functions, for which we prove an upper bound and a lower bound on the nonlinearity, seems also interesting. The fifth class includes functions with an optimal algebraic degree, good nonlinearity and good AI.
Probabilistic rules “if A then B” rules are typically formalized as Bayesian conditionals P (B|A), as many (e.g., Pearl) have argued that Bayesian conditionals are the correct way to think about such rules. However, there are challenges with standard inferences such as modus ponens, modus tollens, and rule chaining that might make probabilistic material implication a better candidate at times for rule-based systems employing forward-chaining; and arguably material implication is still suitable when information about prior or conditional probabilities is not available. We compare a probabilistic version of the material conditional from classical logic with Bayesian conditionals in the setting of interval-valued probability theory. We give quantitative treatments of familiar rules from logic to arrive at the best possible bounds on inferences in rule-based systems constituting directed acyclic graphs. Introduction. The discussion of how to best render formally the intuitions behind “if-then” conditionals is still unresolved, in part because such conditionals can be used to capture analytical truths (“if x is an even number, x is divisiable by two”), conceptual relationships (“if x is a human, then x is mortal”), inductive inferences (“if the sprinkler is on, the grass will become wet”), abductive inferences (“if the light switch is on but the light is off, the light bulb is broken”), normative constraints (“if the traffic light is red, you are not allowed to drive”), hypotheticals (“if I were to run fast, I would be out of breath”), and counterfactuals (“if the driver had been able to break in time, they would not have killed the deer”). From a logical point of view, the classical conditional A → B is false only when A is true and B is false, thus A and A → B cannot both be false at the same time, a fact that can be used to determine the consistency of a set of rules. Yet, treating conditionals as truth-functional has long been seen to be problematic (e.g., Adams 1965) and the widely accepted solution is to view such rules as probabilistic and best modeled by the Bayesian conditional P (B|A) (e.g., Pearl 1988). At the same time, some have argued that under certain conditions the intuitive conditionals amounts to material implication (see Khoo and Mandelkern 2018, for a discussion) and that the Bayesian conditional is not appropriate for handling (indicative) conditionals. Moreover, some logically valid inferences such as the “hypothetical syllogism” A → B,B → C |= A → C (e.g., Hailperin 1984) are often probabilistically “uninformative” (e.g., Pfeifer and Kleiter 2006), as we will also show below. The question then is how probabilistic versions of frequently used conditional inference schemes such as modus ponens, modus tollens, and hypothetical syllogism (used for chaining conditionals) compare to a probabilistic version of the material conditional. The goal of this paper is to shed light on the tradeoffs between the two interpretations of conditionals with respect to the various inference schemes. For this purpose, we use the standard embedding of propositional logic in probability theory, probability logic (e.g., Hailperin 1984), to compare probabilistic bounds on inferences with material conditionals vs. Bayesian conditionals and show that probabilistic material conditionals allow for better and faster inferences with a probabilistic conditional knowledge base. Moreover, we prove the best possible bounds on inferences in directed acyclic graphs induced by a system of probabilistic material conditionals.
Osteoarthritis is one of the most common diseases among elderly people worldwide. This disease affects bone joints causing the movement of the joint very painful. Proper and timely diagnosis is the key in controlling the disease. This study deals with knee osteoarthritis X-ray images for disease detection purpose. As the disease severity increases, the knee joint space gets reduced and the associated bone shapes change. Based on this femur and tibia shape change and knee joint space narrowing, presence of osteoarthritis can be detected. This change of appearance can be easily identified in higher severity levels, but in case of early osteoarthritis the changes are minimal. This makes detection of lower grade osteoarthritis from the healthy ones very difficult. The dataset used here is taken from a publicly available dataset named Knee Osteoarthritis Severity Grading Dataset (Mendeley Data) which is organized from well known OAI dataset. For the purpose training phase of osteoarthritis detection, the data has been divided into two classes having 3332 healthy and 2446 diseased X-ray images. Since changes caused by the disease is not visible in X-ray properly until grade 2 of the disease as per medical literature, this study considers the grade 0 and grade 1 osteoarthritis as healthy group while grade 2, 3 and 4 are considered as parts of diseased group. Pre-processing activities have been carried out on the X-ray images based on pixel intensity profiles. Disease detection has been done here using a light-weight CNN based on MobileNet V1. Lesser number of layers have been used in the proposed approach. Depth-wise convolution has been done in each of the layer which makes the whole process computationally inexpensive. CONTACT Sabyasachi Mukherjee Email: sabya84@gmail.com CONTACT Oishila Bandyopadhyay Email: oishila@gmail.com
