Rewriting is a formalism widely used in computer science and mathematical logic. The classical formalism has been extended, in the context of functional languages, with an order over the rules and, in the context of rewrite based languages, with the negation over patterns. We propose in this paper a concise and clear algorithm computing the difference over patterns which can be used to define generic encodings of constructor term rewriting systems with negation and order into classical term rewriting systems. As a direct consequence, established methods used for term rewriting systems can be applied to analyze properties of the extended systems. The approach can also be seen as a generic compiler which targets any language providing basic pattern matching primitives. The formalism provides also a new method for deciding if a set of patterns subsumes a given pattern and thus, for checking the presence of useless patterns or the completeness of a set of patterns.
L. Maranget has proposed an algorithm for detecting useless patterns @cite_8 for OCaml and Haskell. As mentioned previously, the algorithm in Figure can be also used to check whether a pattern is useless a set of patterns (Proposition ) but it computes the difference between the pattern and the set in order to make the decision. The minimization algorithm in Figure can thus use the two algorithms interchangeably. Both algorithms have been implemented and we measured the execution time for the minimization function on various examples. In average, L. Maranget's approach is @math (l) ! !r, (x l) ! !r', @math @math [ (l) r (x) r' ] @math 20 $25 using strategies, the order of rule application is expressed with a left-to-right strategy choice-operator and for such strategies the gain with our new approach is even more significant than for the examples in @cite_6 which involved only plain, order independent, TRSs.
@cite_6 @cite_8
2056860348 2060814785 2962959718 2114147096
In this paper we present a Bayesian framework for parsing images into their constituent visual patterns. The parsing algorithm optimizes the posterior probability and outputs a scene representation as a "parsing graph", in a spirit similar to parsing sentences in speech and natural language. The algorithm constructs the parsing graph and re-configures it dynamically using a set of moves, which are mostly reversible Markov chain jumps. This computational framework integrates two popular inference approaches--generative (top-down) methods and discriminative (bottom-up) methods. The former formulates the posterior probability in terms of generative models for images defined by likelihood functions and priors. The latter computes discriminative probabilities based on a sequence (cascade) of bottom-up tests filters. In our Markov chain algorithm design, the posterior probability, defined by the generative models, is the invariant (target) probability for the Markov chain, and the discriminative probabilities are used to construct proposal probabilities to drive the Markov chain. Intuitively, the bottom-up discriminative probabilities activate top-down generative models. In this paper, we focus on two types of visual patterns--generic visual patterns, such as texture and shading, and object patterns including human faces and text. These types of patterns compete and cooperate to explain the image and so image parsing unifies image segmentation, object detection, and recognition (if we use generic visual patterns only then image parsing will correspond to image segmentation (Tu and Zhu, 2002. IEEE Trans. PAMI, 24(5):657--673). We illustrate our algorithm on natural images of complex city scenes and show examples where image segmentation can be improved by allowing object specific knowledge to disambiguate low-level segmentation cues, and conversely where object detection can be improved by using generic visual patterns to explain away shadows and occlusions. Recently sparse representation has been applied to visual tracker by modeling the target appearance using a sparse approximation over a template set, which leads to the so-called L1 trackers as it needs to solve an l 1 norm related minimization problem for many times. While these L1 trackers showed impressive tracking accuracies, they are very computationally demanding and the speed bottleneck is the solver to l 1 norm minimizations. This paper aims at developing an L1 tracker that not only runs in real time but also enjoys better robustness than other L1 trackers. In our proposed L1 tracker, a new l 1 norm related minimization model is proposed to improve the tracking accuracy by adding an l 1 norm regularization on the coefficients associated with the trivial templates. Moreover, based on the accelerated proximal gradient approach, a very fast numerical solver is developed to solve the resulting l 1 norm related minimization problem with guaranteed quadratic convergence. The great running time efficiency and tracking accuracy of the proposed tracker is validated with a comprehensive evaluation involving eight challenging sequences and five alternative state-of-the-art trackers. The idea that many important classes of signals can be well-represented by linear combinations of a small set of atoms selected from a given dictionary has had dramatic impact on the theory and practice of signal processing. For practical problems in which an appropriate sparsifying dictionary is not known ahead of time, a very popular and successful heuristic is to search for a dictionary that minimizes an appropriate sparsity surrogate over a given set of sample data. While this idea is appealing, the behavior of these algorithms is largely a mystery; although there is a body of empirical evidence suggesting they do learn very eective representations, there is little theory to guarantee when they will behave correctly, or when the learned dictionary can be expected to generalize. In this paper, we take a step towards such a theory. We show that under mild hypotheses, the dictionary learning problem is locally well-posed: the desired solution is indeed a local minimum of the ‘ 1 norm. Namely, if A 2 R m n is an incoherent (and possibly overcomplete) dictionary, and the coecients X 2 R n p follow a random sparse model, then with high probability (A;X) is a local minimum of the ‘ 1 norm over the manifold of factorizations (A 0 ;X 0 ) satisfying A 0 X 0 = Y , provided the number of samples p = We consider inexact linear equations y ≈ Φx where y is a given vector in R n , Φ is a given n x m matrix, and we wish to find x 0,∈ as sparse as possible while obeying ∥y - Φx 0,∈ ∥ 2 ≤ ∈. In general, this requires combinatorial optimization and so is considered intractable. On the other hand, the l 1 -minimization problem min ∥x∥ 1 subject to ∥y - Φx∥ 2 ≤ e is convex and is considered tractable. We show that for most Φ, if the optimally sparse approximation x 0,∈ is sufficiently sparse, then the solution x 1,∈ of the l 1 -minimization problem is a good approximation to x 0,∈ . We suppose that the columns of Φ are normalized to the unit l 2 -norm, and we place uniform measure on such Φ. We study the underdetermined case where m ∼ τn and τ > 1, and prove the existence of p = p(r) > 0 and C = C(p, τ) so that for large n and for all Φ's except a negligible fraction, the following approximate sparse solution property of Φ holds: for every y having an approximation ∥y - Φx 0 ∥ 2 ≤ ∈ by a coefficient vector x 0 e R m with fewer than ρ · n nonzeros, ∥x 1,∈ - x 0 ∥ 2 ≤ C ≤ ∈. This has two implications. First, for most Φ, whenever the combinatorial optimization result x 0,∈ would be very sparse, x 1,∈ is a good approximation to x 0,∈ . Second, suppose we are given noisy data obeying y = Φx 0 + z where the unknown x 0 is known to be sparse and the noise ∥z∥ 2 ≤ ∈. For most Φ, noise-tolerant l 1 -minimization will stably recover x 0 from y in the presence of noise z. We also study the barely determined case m = n and reach parallel conclusions by slightly different arguments. Proof techniques include the use of almost-spherical sections in Banach space theory and concentration of measure for eigenvalues of random matrices.