Annals of Mathematics and Artificial Intelligence最新文献

Topological representations of binary digital images usually take into consideration different adjacency types between colors. Within the cubical-voxel 3D binary image context, we design an algorithm for computing the isotopic model of an image, called (6, 26)-Homological Region Adjacency Tree ((6, 26)-Hom-Tree). This algorithm is based on a flexible graph scaffolding at the inter-voxel level called Homological Spanning Forest model (HSF). Hom-Trees are edge-weighted trees in which each node is a maximally connected set of constant-value voxels, which is interpreted as a subtree of the HSF. This representation integrates and relates the homological information (connected components, tunnels and cavities) of the maximally connected regions of constant color using 6-adjacency and 26-adjacency for black and white voxels, respectively (the criteria most commonly used for 3D images). The Euler-Poincaré numbers (which may as well be computed by counting the number of cells of each dimension on a cubical complex) and the connected component labeling of the foreground and background of a given image can also be straightforwardly computed from its Hom-Trees. Being (I_D) a 3D binary well-composed image (where D is the set of black voxels), an almost fully parallel algorithm for constructing the Hom-Tree via HSF computation is implemented and tested here. If (I_D) has (m_1{times } m_2{times } m_3) voxels, the time complexity order of the reproducible algorithm is near (O(log (m_1{+}m_2{+}m_3))), under the assumption that a processing element is available for each cubical voxel. Strategies for using the compressed information of the Hom-Tree representation to distinguish two topologically different images having the same homological information (Betti numbers) are discussed here. The topological discriminatory power of the Hom-Tree and the low time complexity order of the proposed implementation guarantee its usability within machine learning methods for the classification and comparison of natural 3D images.

We study the stability of accuracy during the training of deep neural networks (DNNs). In this context, the training of a DNN is performed via the minimization of a cross-entropy loss function, and the performance metric is accuracy (the proportion of objects that are classified correctly). While training results in a decrease of loss, the accuracy does not necessarily increase during the process and may sometimes even decrease. The goal of achieving stability of accuracy is to ensure that if accuracy is high at some initial time, it remains high throughout training. A recent result by Berlyand, Jabin, and Safsten introduces a doubling condition on the training data, which ensures the stability of accuracy during training for DNNs using the absolute value activation function. For training data in (mathbb {R}^n), this doubling condition is formulated using slabs in (mathbb {R}^n) and depends on the choice of the slabs. The goal of this paper is twofold. First, to make the doubling condition uniform, that is, independent of the choice of slabs. This leads to sufficient conditions for stability in terms of training data only. In other words, for a training set T that satisfies the uniform doubling condition, there exists a family of DNNs such that a DNN from this family with high accuracy on the training set at some training time (t_0) will have high accuracy for all time (t>t_0). Moreover, establishing uniformity is necessary for the numerical implementation of the doubling condition. We demonstrate how to numerically implement a simplified version of this uniform doubling condition on a dataset and apply it to achieve stability of accuracy using a few model examples. The second goal is to extend the original stability results from the absolute value activation function to a broader class of piecewise linear activation functions with finitely many critical points, such as the popular Leaky ReLU.

We study a problem of best-effort adaptation motivated by several applications and considerations, which consists of determining an accurate predictor for a target domain, for which a moderate amount of labeled samples are available, while leveraging information from another domain for which substantially more labeled samples are at one’s disposal. We present a new and general discrepancy-based theoretical analysis of sample reweighting methods, including bounds holding uniformly over the weights. We show how these bounds can guide the design of learning algorithms that we discuss in detail. We further show that our learning guarantees and algorithms provide improved solutions for standard domain adaptation problems, for which few labeled data or none are available from the target domain. We finally report the results of a series of experiments demonstrating the effectiveness of our best-effort adaptation and domain adaptation algorithms, as well as comparisons with several baselines. We also discuss how our analysis can benefit the design of principled solutions for fine-tuning.

In this paper, we consider three Relaxation Adaptive Memory Programming (RAMP) approaches for solving the Uncapacitated Facility Location Problem (UFLP), whose objective is to locate a set of facilities and allocate these facilities to all clients at minimum cost. Different levels of sophistication were implemented to measure the performance of the RAMP approach. In the simpler level, (Dual-) RAMP explores more intensively the dual side of the problem, incorporating a Lagrangean Relaxation and Subgradient Optimization with a simple Improvement Method on the primal side. In the most sophisticated level, RAMP combines a Dual-Ascent procedure on the dual side with a Scatter Search (SS) procedure on primal side, forming the Primal–Dual RAMP (PD-RAMP). The Dual-RAMP algorithm starts with (dual side) the dualization of the initial problem, and then a projection method projects the dual solutions into the primal solutions space. Next, (primal side) the projected solutions are improved through an improvement method. In the PD-RAMP algorithm, the SS procedure is incorporated in the primal side to carry out a more intensive exploration. The algorithm alternates between the dual and the primal side until a fixed number of iterations is achieved. Computational experiments on a standard testbed for the UFLP were conducted to assess the performance of all the RAMP algorithms.

A switch-list representation (SLR) of a Boolean function is a compressed truth table representation of a Boolean function in which only (i) the function value of the first row in the truth table and (ii) a list of switches are stored. A switch is a Boolean vector whose function value differs from the value of the preceding Boolean vector in the truth table. The paper Čepek and Chromý (JAIR 2020) systematically studies the properties of SLRs and among other results gives polynomial-time algorithms for all standard queries investigated in the Knowledge Compilation Map introduced in Darwiche and Marquis (JAIR 2002). These queries include consistency check, validity check, clausal entailment check, implicant check, equivalence check, sentential entailment check, model counting, and model enumeration. The most difficult query supported in polynomial time by the smallest number of representation languages considered in the Knowledge Compilation Map is the sentential entailment check (of which the equivalence check is a special case). This query can be answered in polynomial time for SLRs, as shown in Čepek and Chromý (JAIR 2020). However, the query-answering algorithm is an indirect one: it first compiles both input SLRs into OBDDs (changing the order of variables for one of them if necessary) and then runs the sentential entailment check on the constructed OBDDs (both respecting the same order of variables) using an algorithm from the monograph by Wegener (2000). In this paper we present algorithms that answer both the equivalence and the sentential entailment query directly by manipulating the input SLRs (hence eliminating the compilation step into OBDD), which in both cases improves the time complexity of answering the query by a factor of n for input SLRs on n variables.

Search/optimization problems are plentiful in scientific and engineering domains. Artificial intelligence has long contributed to the development of search algorithms and declarative programming languages geared toward solving and modeling search/optimization problems. Automated reasoning and knowledge representation are the subfields of AI that are particularly vested in these developments. Many popular automated reasoning paradigms provide users with languages supporting optimization statements: answer set programming or MaxSAT or min-one, to name a few. These paradigms vary significantly in their languages and in the ways they express quality conditions on computed solutions. Here we propose a unifying framework of so-called weight systems that eliminates syntactic distinctions between paradigms and allows us to see essential similarities and differences between optimization statements provided by paradigms. This unifying outlook has significant simplifying and explanatory potential in the studies of optimization and modularity in automated reasoning and knowledge representation. It also supplies researchers with a convenient tool for proving the formal properties of distinct frameworks; bridging these frameworks; and facilitating the development of translational solvers.