Matrix profile has been recently proposed as a promising technique to the problem of all-pairs-similarity search on time series. Efficient algorithms have been proposed for computing it, e.g., STAMP, STOMP and SCRIMP++. All these algorithms use the z-normalized Euclidean distance to measure the distance between subsequences. However, as we observed, for some datasets other Euclidean measurements are more useful for knowledge discovery from time series. In this paper, we propose efficient algorithms for computing matrix profile for a general class of Euclidean distances. We first propose a simple but efficient algorithm called AAMP for computing matrix profile with the "pure" (non-normalized) Euclidean distance. Then, we extend our algorithm for the p-norm distance. We also propose an algorithm, called ACAMP, that uses the same principle as AAMP, but for the case of z-normalized Euclidean distance. We implemented our algorithms, and evaluated their performance through experimentation. The experiments show excellent performance results. For example, they show that AAMP is very efficient for computing matrix profile for non-normalized Euclidean distances. The results also show that the ACAMP algorithm is significantly faster than SCRIMP++ (the state of the art matrix profile algorithm) for the case of z-normalized Euclidean distance.
In @cite_9 , proposed an algorithm, called STOMP, that is faster than STAMP. The STOMP algorithm is similar to STAMP in that it can be seen as highly optimized nested loop searches, with the repeated calculation of distance profiles as the inner loop. However, while STAMP must evaluate the distance profiles in random order (to allow its anytime behavior), STOMP performs an ordered search. STOMP exploits the locality of these searches, and reduces the time complexity by a factor of @math . In @cite_0 , the authors proposed an extension of STOMP, called SCRIMP++, that converges much faster than STOMP. To the best of our knowledge, all most all matrix profile algorithms have been developed for z-normalized Euclidean distance. In this paper, we proposed efficient algorithms for a larger class of Euclidean functions. We also proposed an algorithm for the z-normalized case, , ACAMP, that is significantly faster than SCRIMP++, which is the fastest exact algorithm for matrix profile computation in the literature, to the best of our knowledge. Our ACAMP algorithm is designed based on an efficient incremental technique that does not need to calculate FFT (in contrast to SCRIMP++).
@cite_0 @cite_9
2069278600 2951481254 2141653564 1964470356
We present a new efficient algorithm for the search version of the approximate Closest Vector Problem with Preprocessing (CVPP). Our algorithm achieves an approximation factor of O(n sqrt log n ), improving on the previous best of O(n^ 1.5 ) due to Lag arias, Lenstra, and Schnorr hkzbabai . We also show, somewhat surprisingly, that only O(n) vectors of preprocessing advice are sufficient to solve the problem (with the slightly worse approximation factor of O(n)). We remark that this still leaves a large gap with respect to the decisional version of CVPP, where the best known approximation factor is O(sqrt n log n ) due to Aharonov and Regev AharonovR04 . To achieve these results, we show a reduction to the same problem restricted to target points that are close to the lattice and a more efficient reduction to a harder problem, Bounded Distance Decoding with preprocessing (BDDP). Combining either reduction with the previous best-known algorithm for BDDP by Liu, Lyubashevsky, and Micciancio LiuLM06 gives our main result. In the setting of CVP without preprocessing, we also give a reduction from (1+eps)gamma approximate CVP to gamma approximate CVP where the target is at distance at most 1+1 eps times the minimum distance (the length of the shortest non-zero vector) which relies on the lattice sparsification techniques of Dadush and Kun DadushK13 . As our final and most technical contribution, we present a substantially more efficient variant of the LLM algorithm (both in terms of run-time and amount of preprocessing advice), and via an improved analysis, show that it can decode up to a distance proportional to the reciprocal of the smoothing parameter of the dual lattice MR04 . We show that this is never smaller than the LLM decoding radius, and that it can be up to an wide tilde Omega (sqrt n ) factor larger. In this paper, we propose a new stochastic alternating direction method of multipliers (ADMM) algorithm, which incrementally approximates the full gradient in the linearized ADMM formulation. Besides having a low per-iteration complexity as existing stochastic ADMM algorithms, the proposed algorithm improves the convergence rate on convex problems from @math to @math , where @math is the number of iterations. This matches the convergence rate of the batch ADMM algorithm, but without the need to visit all the samples in each iteration. Experiments on the graph-guided fused lasso demonstrate that the new algorithm is significantly faster than state-of-the-art stochastic and batch ADMM algorithms. The decremental single-source shortest paths (SSSP) problem concerns maintaining the distances between a given source node s to every node in an n-node m-edge graph G undergoing edge deletions. While its static counterpart can be easily solved in near-linear time, this decremental problem is much more challenging even in the undirected unweighted case. In this case, the classic O(mn) total update time of Even and Shiloach (JACM 1981) has been the fastest known algorithm for three decades. With the loss of a (1 + aepsi;)-approximation factor, the running time was recently improved to O(n2+o(1)) by Bernstein and Roditty (SODA 2011), and more recently to O(n1.8+o(1) + m1+o(1)) by Henzinger, Krinninger, and Nanongkai (SODA 2014). In this paper, we finally bring the running time of this case down to near-linear: We give a (1 + aepsi;)-approximation algorithm with O(m1+o(1)) total update time, thus obtaining near-linear time. Moreover, we obtain O(m1+o(1)log W) time for the weighted case, where the edge weights are integers from 1 to W. The only prior work on weighted graphs in o(mn log W) time is the O(mn0.986 log W)-time algorithm by Henzinger, Krinninger, and Nanongkai (STOC 2014) which works for the general weighted directed case. In contrast to the previous results which rely on maintaining a sparse emulator, our algorithm relies on maintaining a so-called sparse (d, aepsi;)-hop set introduced by Cohen (JACM 2000) in the PRAM literature. A (d, aepsi;)-hop set of a graph G = (V, E) is a set E' of weighted edges such that the distance between any pair of nodes in G can be (1 + aepsi;)-approximated by their d-hop distance (given by a path containing at most d edges) on G'=(V, E a#x222A; E'). Our algorithm can maintain an (no(1), aepsi;)-hop set of near-linear size in near-linear time under edge deletions. It is the first of its kind to the best of our knowledge. To maintain the distances on this hop set, we develop a monotone bounded-hop Even-Shiloach tree. It results from extending and combining the monotone Even-Shiloach tree of Henzinger, Krinninger, and Nanongkai (FOCS 2013) with the bounded-hop SSSP technique of Bernstein (STOC 2013). These two new tools might be of independent interest. We propose an efficient and robust solution for image set classification. A joint representation of an image set is proposed which includes the image samples of the set and their affine hull model. The model accounts for unseen appearances in the form of affine combinations of sample images. To calculate the between-set distance, we introduce the Sparse Approximated Nearest Point (SANP). SANPs are the nearest points of two image sets such that each point can be sparsely approximated by the image samples of its respective set. This novel sparse formulation enforces sparsity on the sample coefficients and jointly optimizes the nearest points as well as their sparse approximations. Unlike standard sparse coding, the data to be sparsely approximated are not fixed. A convex formulation is proposed to find the optimal SANPs between two sets and the accelerated proximal gradient method is adapted to efficiently solve this optimization. We also derive the kernel extension of the SANP and propose an algorithm for dynamically tuning the RBF kernel parameter while matching each pair of image sets. Comprehensive experiments on the UCSD Honda, CMU MoBo, and YouTube Celebrities face datasets show that our method consistently outperforms the state of the art.