We consider the problem of PAC-learning decision trees, i.e., learning a decision tree over the n-dimensional hypercube from independent random labeled examples. Despite significant effort, no polynomial-time algorithm is known for learning polynomial-sized decision trees (even trees of any super-constant size), even when examples are assumed to be drawn from the uniform distribution on 0,1 ^n. We give an algorithm that learns arbitrary polynomial-sized decision trees for most product distributions . In particular, consider a random product distribution where the bias of each bit is chosen independently and uniformly from, say, [.49,.51]. Then with high probability over the parameters of the product distribution and the random examples drawn from it, the algorithm will learn any tree. More generally, in the spirit of smoothed analysis, we consider an arbitrary product distribution whose parameters are specified only up to a [-c,c] accuracy (perturbation), for an arbitrarily small positive constant c.
We consider a smoothed learning'' model inspired by Smoothed Analysis, which Spielman and Teng introduced to explain why the simplex method for linear programming (LP) usually runs in polynomial time @cite_9 . Roughly speaking, they show that if each parameter of an LP is perturbed by a small amount, then the simplex method will run in polynomial time with high probability (in fact, the expected run-time will be polynomial). For LP's arising from nature or business (as opposed to reduction from another computational problem), the parameters are measurements or estimates that have some inherent inaccuracy or uncertainty. Hence, the model is reasonable for a large class of interesting LP's.
@cite_9
2323401846 2161444532 789250018 2127459401
We describe an end-to-end framework for learning parameters of min-cost flow multitarget tracking problem with quadratic trajectory interactions including suppression of overlapping tracks and contextual cues about co-occurrence of different objects. Our approach utilizes structured prediction with a tracking-specific loss function to learn the complete set of model parameters. Under our learning framework, we evaluate two different approaches to finding an optimal set of tracks under quadratic model objective based on an LP relaxation and a novel greedy extension to dynamic programming that handles pairwise interactions. We find the greedy algorithm achieves almost equivalent accuracy to the LP relaxation while being 2-7x faster than a commercial solver. We evaluate trained models on the challenging MOT and KITTI benchmarks. Surprisingly, we find that with proper parameter learning, our simple data-association model without explicit appearance motion reasoning is able to outperform many state-of-the-art methods that use far more complex motion features and affinity metric learning. A linear programming (LP) approach is proposed for the weighted graph matching problem. A linear program is obtained by formulating the graph matching problem in L sub 1 norm and then transforming the resulting quadratic optimization problem to a linear one. The linear program is solved using a simplex-based algorithm. Then, approximate 0-1 integer solutions are obtained by applying the Hungarian method on the real solutions of the linear program. The complexity of the proposed algorithm is polynomial time, and it is O(n sup 6 L) for matching graphs of size n. The developed algorithm is compared to two other algorithms. One is based on an eigendecomposition approach and the other on a symmetric polynomial transform. Experimental results showed that the LP approach is superior in matching graphs than both other methods. > Time series data mining has received much greater interest along with the increase in temporal data sets from different domains such as medicine, finance, multimedia, etc. Representations are important to reduce dimensionality and generate useful similarity measures. High-level representations such as Fourier transforms, wavelets, piecewise polynomial models, etc., were considered previously. Recently, autoregressive kernels were introduced to reflect the similarity of the time series. We introduce a novel approach to model the dependency structure in time series that generalizes the concept of autoregression to local autopatterns. Our approach generates a pattern-based representation along with a similarity measure called learned pattern similarity (LPS). A tree-based ensemble-learning strategy that is fast and insensitive to parameter settings is the basis for the approach. Then, a robust similarity measure based on the learned patterns is presented. This unsupervised approach to represent and measure the similarity between time series generally applies to a number of data mining tasks (e.g., clustering, anomaly detection, classification). Furthermore, an embedded learning of the representation avoids pre-defined features and an extraction step which is common in some feature-based approaches. The method generalizes in a straightforward manner to multivariate time series. The effectiveness of LPS is evaluated on time series classification problems from various domains. We compare LPS to eleven well-known similarity measures. Our experimental results show that LPS provides fast and competitive results on benchmark datasets from several domains. Furthermore, LPS provides a research direction and template approach that breaks from the linear dependency models to potentially foster other promising nonlinear approaches. We consider supervised learning problems within the positive-definite kernel framework, such as kernel ridge regression, kernel logistic regression or the support vector machine. With kernels leading to infinite-dimensional feature spaces, a common practical limiting difficulty is the necessity of computing the kernel matrix, which most frequently leads to algorithms with running time at least quadratic in the number of observations n, i.e., O(n^2). Low-rank approximations of the kernel matrix are often considered as they allow the reduction of running time complexities to O(p^2 n), where p is the rank of the approximation. The practicality of such methods thus depends on the required rank p. In this paper, we show that in the context of kernel ridge regression, for approximations based on a random subset of columns of the original kernel matrix, the rank p may be chosen to be linear in the degrees of freedom associated with the problem, a quantity which is classically used in the statistical analysis of such methods, and is often seen as the implicit number of parameters of non-parametric estimators. This result enables simple algorithms that have sub-quadratic running time complexity, but provably exhibit the same predictive performance than existing algorithms, for any given problem instance, and not only for worst-case situations.