Many natural signals exhibit a sparse representation, whenever a suitable describing model is given. Here, a linear generative model is considered, where many sparsity-based signal processing techniques rely on such a simplified model. As this model is often unknown for many classes of the signals, we need to select such a model based on the domain knowledge or using some exemplar signals. This paper presents a new exemplar based approach for the linear model (called the dictionary) selection, for such sparse inverse problems. The problem of dictionary selection, which has also been called the dictionary learning in this setting, is first reformulated as a joint sparsity model. The joint sparsity model here differs from the standard joint sparsity model as it considers an overcompleteness in the representation of each signal, within the range of selected subspaces. The new dictionary selection paradigm is examined with some synthetic and realistic simulations.

While several papers have investigated computationally and statistically efficient methods for learning Gaussian mixtures, precise minimax bounds for their statistical performance as well as fundamental limits in high-dimensional settings are not well-understood. In this paper, we provide precise information theoretic bounds on the clustering accuracy and sample complexity of learning a mixture of two isotropic Gaussians in high dimensions under small mean separation. If there is a sparse subset of relevant dimensions that determine the mean separation, then the sample complexity only depends on the number of relevant dimensions and mean separation, and can be achieved by a simple computationally efficient procedure. Our results provide the first step of a theoretical basis for recent methods that combine feature selection and clustering.

Temporal-difference (TD) learning is one of the most successful and broadly applied solutions to the reinforcement learning problem; it has been used to achieve master-level play in chess, checkers and backgammon. Monte-Carlo tree search is a recent algorithm for simulation-based search, which has been used to achieve master-level play in Go. We have introduced a new approach to high-performance planning. Our method, TD search, combines TD learning with simulation-based search. Like Monte-Carlo tree search, value estimates are updated by learning online from simulated experience. Like TD learning, it uses value function approximation and bootstrapping to efficiently generalise between related states. We applied TD search to the game of 9x9 Go, using a million binary features matching simple patterns of stones. Without any explicit search tree, our approach outperformed a vanilla Monte-Carlo tree search with the same number of simulations. When combined with a simple alpha-beta search, our program also outperformed all traditional (pre-Monte-Carlo) search and machine learning programs on the 9x9 Computer Go Server.

Perimeter search is a bidirectional search algorithm consisting of two phases. In the first phase, a limited regression search computes the perimeter, a region which must necessarily be passed in every solution. In the second phase, a heuristic forward search finds an optimal plan from the initial state to the perimeter. The drawback of perimeter search is the need to compute heuristic estimates towards every state on the perimeter in the forward phase. We show that this limitation can be effectively overcome when using pattern database (PDB) heuristics in the forward phase. The combination of perimeter search and PDB heuristics has been considered previously by Felner and Ofek for solving combinatorial puzzles. They claimed that, based on theoretical considerations and experimental evidence, the use of perimeter search in this context offers "limited or no benefits". Our theoretical and experimental results show that this assessment should be revisited.

Most bounded suboptimal algorithms in the search literature have been developed so as to be epsilon-admissible. This means that the solutions found by these algorithms are guaranteed to be no more than a factor of (1 + ε) greater than optimal. However, this is not the only possible form of suboptimality bounding. For example, another possible suboptimality guarantee is that of additive bounding, which requires that the cost of the solution found is no more than the cost of the optimal solution plus a constant gamma. In this work, we consider the problem of developing algorithms so as to satisfy a given, and arbitrary, suboptimality requirement. To do so, we develop a theoretical framework which can be used to construct algorithms for a large class of possible suboptimality paradigms. We then use the framework to develop additively bounded algorithms, and show that in practice these new algorithms effectively trade-off additive solution suboptimality for runtime.

In heuristic search and especially in optimal classical planning the computation of accurate heuristic values can take up the majority of runtime. In many cases, the heuristic computations for a search node and its successors are very similar, leading to significant duplication of effort. For example most landmarks of a node that are computed by the LM-cut algorithm are also landmarks for the node's successors. We propose to reuse these landmarks and incrementally compute new ones to speed up the LM-cut calculation. The speed advantage obtained by incremental computation is offset by higher memory usage. We investigate different search algorithms that reduce memory usage without sacrificing the faster computation, leading to a substantial increase in coverage for benchmark domains from the International Planning Competitions.

Dynamic programming is a well-known approach for solving MDPs. In large state spaces, asynchronous versions like Real-Time Dynamic Programming have been applied successfully. If unfolded into equivalent trees, Monte-Carlo Tree Search algorithms are a valid alternative. UCT, the most popular representative, obtains good anytime behavior by guiding the search towards promising areas of the search tree. The Heuristic Search algorithm AO∗ finds optimal solutions for MDPs that can be represented as acyclic AND/OR graphs. We introduce a common framework, Trial-based Heuristic Tree Search, that subsumes these approaches and distinguishes them based on five ingredients: heuristic function, backup function, action selection, outcome selection, and trial length. Using this framework, we describe three new algorithms which mix these ingredients in novel ways in an attempt to combine their different strengths. Our evaluation shows that two of our algorithms not only provide superior theoretical properties to UCT, but also outperform state-of-the-art approaches experimentally.

Path planning is often a high-dimensional computationally-expensive planning problem as it requires reasoning about the kinodynamic constraints of the robot and collisions of the robot with the environment. However, large regions of the environment are typically benign enough that a much faster low-dimensional planning combined with a local path following controller suffice. Planning with Adaptive Dimensionality that was recently developed makes use of this observation and iteratively constructs and searches a state-space consisting of mainly low-dimensional states. It only introduces regions of high-dimensional states into the state-space where they are necessary to ensure completeness and bounds on sub-optimality. However, due to its iterative nature, the approach relies on running a series of weighted A* searches. In this paper, we introduce and apply to Planning with Adaptive Dimensionality a simple but very effective incremental version of weighted A* that reuses its previously generated search tree if available. On the theoretical side, the new algorithm preserves guarantees on completeness and bounds on sub-optimality. On the experimental side, it speeds up 3D (x,y,heading) path planning with a full-body collision checking by up to a factor of 5. Our results also show that it tends to be much faster than applying alternative incremental graph search techniques such as D* to Planning with Adaptive Dimensionality.

Many real-world planning domains, including those used in common benchmark problems, are based on multiagent scenarios. It has long been recognised that breaking down such problems into sub-problems for individual agents may help reduce overall planning complexity. This kind of approach is especially effective in domains where interaction between agents is limited. In this paper we present a fully centralised, offline, sequential, total-order planning algorithm for solving classical planning problems based on this idea. This algorithm consists of an automated decomposition process and a heuristic search method designed specifically for decomposed domains. The decomposition method is part of a preprocessing step and can be used to determine the 'multiagent nature' of a planning problem prior to actual plan search. The heuristic search strategy is shown to effectively exploit any decompositions that are found and performs significantly better than current approaches on loosely coupled domains.

We describe the Fast Greedy Sparse Subspace Clustering (FGSSC) algorithm providing an efficient method for clustering data belonging to a few low-dimensional linear or affine subspaces. The main difference of our algorithm from predecessors is its ability to work with noisy data having a high rate of erasures (missed entries with the known coordinates) and errors (corrupted entries with unknown coordinates). We discuss here how to implement the fast version of the greedy algorithm with the maximum efficiency whose greedy strategy is incorporated into iterations of the basic algorithm. We provide numerical evidences that, in the subspace clustering capability, the fast greedy algorithm outperforms not only the existing state-of-the art SSC algorithm taken by the authors as a basic algorithm but also the recent GSSC algorithm. At the same time, its computational cost is only slightly higher than the cost of SSC. The numerical evidence of the algorithm significant advantage is presented for a few synthetic models as well as for the Extended Yale B dataset of facial images. In particular, the face recognition misclassification rate turned out to be 6-20 times lower than for the SSC algorithm. We provide also the numerical evidence that the FGSSC algorithm is able to perform clustering of corrupted data efficiently even when the sum of subspace dimensions significantly exceeds the dimension of the ambient space.