Archetypal analysis is an unsupervised learning method for exploratory data analysis. One major challenge that limits the applicability of archetypal analysis in practice is the inherent computational complexity of the existing algorithms. In this paper, we provide a novel approximation approach to partially address this issue. Utilizing probabilistic ideas from high-dimensional geometry, we introduce two preprocessing techniques to reduce the dimension and representation cardinality of the data, respectively. We prove that, provided the data is approximately embedded in a low-dimensional linear subspace and the convex hull of the corresponding representations is well approximated by a polytope with a few vertices, our method can effectively reduce the scaling of archetypal analysis. Moreover, the solution of the reduced problem is near-optimal in terms of prediction errors. Our approach can be combined with other acceleration techniques to further mitigate the intrinsic complexity of archetypal analysis. We demonstrate the usefulness of our results by applying our method to summarize several moderately large-scale datasets.
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We introduce and analyze various Regularized Combined Field Integral Equations (CFIER) formulations of time-harmonic Navier equations in media with piece-wise constant material properties. These formulations can be derived systematically starting from suitable coercive approximations of Dirichlet-to-Neumann operators (DtN), and we present a periodic pseudodifferential calculus framework within which the well posedness of CIER formulations can be established. We also use the DtN approximations to derive and analyze Optimized Schwarz (OS) methods for the solution of elastodynamics transmission problems. The pseudodifferential calculus we develop in this paper relies on careful singularity splittings of the kernels of Navier boundary integral operators which is also the basis of high-order Nystr\"om quadratures for their discretizations. Based on these high-order discretizations we investigate the rate of convergence of iterative solvers applied to CFIER and OS formulations of scattering and transmission problems. We present a variety of numerical results that illustrate that the CFIER methodology leads to important computational savings over the classical CFIE one, whenever iterative solvers are used for the solution of the ensuing discretized boundary integral equations. Finally, we show that the OS methods are competitive in the high-frequency high-contrast regime.
Approximate linear programs (ALPs) are well-known models based on value function approximations (VFAs) to obtain policies and lower bounds on the optimal policy cost of discounted-cost Markov decision processes (MDPs). Formulating an ALP requires (i) basis functions, the linear combination of which defines the VFA, and (ii) a state-relevance distribution, which determines the relative importance of different states in the ALP objective for the purpose of minimizing VFA error. Both these choices are typically heuristic: basis function selection relies on domain knowledge while the state-relevance distribution is specified using the frequency of states visited by a heuristic policy. We propose a self-guided sequence of ALPs that embeds random basis functions obtained via inexpensive sampling and uses the known VFA from the previous iteration to guide VFA computation in the current iteration. Self-guided ALPs mitigate the need for domain knowledge during basis function selection as well as the impact of the initial choice of the state-relevance distribution, thus significantly reducing the ALP implementation burden. We establish high probability error bounds on the VFAs from this sequence and show that a worst-case measure of policy performance is improved. We find that these favorable implementation and theoretical properties translate to encouraging numerical results on perishable inventory control and options pricing applications, where self-guided ALP policies improve upon policies from problem-specific methods. More broadly, our research takes a meaningful step toward application-agnostic policies and bounds for MDPs.
This paper revisits the temporal difference (TD) learning algorithm for the policy evaluation tasks in reinforcement learning. Typically, the performance of TD(0) and TD($\lambda$) is very sensitive to the choice of stepsizes. Oftentimes, TD(0) suffers from slow convergence. Motivated by the tight link between the TD(0) learning algorithm and the stochastic gradient methods, we develop a provably convergent adaptive projected variant of the TD(0) learning algorithm with linear function approximation that we term AdaTD(0). In contrast to the TD(0), AdaTD(0) is robust or less sensitive to the choice of stepsizes. Analytically, we establish that to reach an $\epsilon$ accuracy, the number of iterations needed is $\tilde{O}(\epsilon^{-2}\ln^4\frac{1}{\epsilon}/\ln^4\frac{1}{\rho})$ in the general case, where $\rho$ represents the speed of the underlying Markov chain converges to the stationary distribution. This implies that the iteration complexity of AdaTD(0) is no worse than that of TD(0) in the worst case. When the stochastic semi-gradients are sparse, we provide theoretical acceleration of AdaTD(0). Going beyond TD(0), we develop an adaptive variant of TD($\lambda$), which is referred to as AdaTD($\lambda$). Empirically, we evaluate the performance of AdaTD(0) and AdaTD($\lambda$) on several standard reinforcement learning tasks, which demonstrate the effectiveness of our new approaches.
We argue that proven exponential upper bounds on runtimes, an established area in classic algorithms, are interesting also in heuristic search and we prove several such results. We show that any of the algorithms randomized local search, Metropolis algorithm, simulated annealing, and (1+1) evolutionary algorithm can optimize any pseudo-Boolean weakly monotonic function under a large set of noise assumptions in a runtime that is at most exponential in the problem dimension~$n$. This drastically extends a previous such result, limited to the (1+1) EA, the LeadingOnes function, and one-bit or bit-wise prior noise with noise probability at most $1/2$, and at the same time simplifies its proof. With the same general argument, among others, we also derive a sub-exponential upper bound for the runtime of the $(1,\lambda)$ evolutionary algorithm on the OneMax problem when the offspring population size $\lambda$ is logarithmic, but below the efficiency threshold. To show that our approach can also deal with non-trivial parent population sizes, we prove an exponential upper bound for the runtime of the mutation-based version of the simple genetic algorithm on the OneMax benchmark, matching a known exponential lower bound.
We characterize the complexity of the lattice decoding problem from a neural network perspective. The notion of Voronoi-reduced basis is introduced to restrict the space of solutions to a binary set. On the one hand, this problem is shown to be equivalent to computing a continuous piecewise linear (CPWL) function restricted to the fundamental parallelotope. On the other hand, it is known that any function computed by a ReLU feed-forward neural network is CPWL. As a result, we count the number of affine pieces in the CPWL decoding function to characterize the complexity of the decoding problem. It is exponential in the space dimension $n$, which induces shallow neural networks of exponential size. For structured lattices we show that folding, a technique equivalent to using a deep neural network, enables to reduce this complexity from exponential in $n$ to polynomial in $n$. Regarding unstructured MIMO lattices, in contrary to dense lattices many pieces in the CPWL decoding function can be neglected for quasi-optimal decoding on the Gaussian channel. This makes the decoding problem easier and it explains why shallow neural networks of reasonable size are more efficient with this category of lattices (in low to moderate dimensions).
We present a means of formulating and solving the well known structure-and-motion problem in computer vision with probabilistic graphical models. We model the unknown camera poses and 3D feature coordinates as well as the observed 2D projections as Gaussian random variables, using sigma point parameterizations to effectively linearize the nonlinear relationships between these variables. Those variables involved in every projection are grouped into a cluster, and we connect the clusters in a cluster graph. Loopy belief propagation is performed over this graph, in an iterative re-initialization and estimation procedure, and we find that our approach shows promise in both simulation and on real-world data. The PGM is easily extendable to include additional parameters or constraints.
We show how to apply Sobol's method of global sensitivity analysis to measure the influence exerted by a set of nodes' evidence on a quantity of interest expressed by a Bayesian network. Our method exploits the network structure so as to transform the problem of Sobol index estimation into that of marginalization inference. This way, we can efficiently compute indices for networks where brute-force or Monte Carlo based estimators for variance-based sensitivity analysis would require millions of costly samples. Moreover, our method gives exact results when exact inference is used, and also supports the case of correlated inputs. The proposed algorithm is inspired by the field of tensor networks, and generalizes earlier tensor sensitivity techniques from the acyclic to the cyclic case. We demonstrate the method on three medium to large Bayesian networks that cover the areas of project risk management and reliability engineering.
UMAP (Uniform Manifold Approximation and Projection) is a novel manifold learning technique for dimension reduction. UMAP is constructed from a theoretical framework based in Riemannian geometry and algebraic topology. The result is a practical scalable algorithm that applies to real world data. The UMAP algorithm is competitive with t-SNE for visualization quality, and arguably preserves more of the global structure with superior run time performance. Furthermore, UMAP has no computational restrictions on embedding dimension, making it viable as a general purpose dimension reduction technique for machine learning.
Network embedding aims to learn low-dimensional representations of nodes in a network, while the network structure and inherent properties are preserved. It has attracted tremendous attention recently due to significant progress in downstream network learning tasks, such as node classification, link prediction, and visualization. However, most existing network embedding methods suffer from the expensive computations due to the large volume of networks. In this paper, we propose a $10\times \sim 100\times$ faster network embedding method, called Progle, by elegantly utilizing the sparsity property of online networks and spectral analysis. In Progle, we first construct a \textit{sparse} proximity matrix and train the network embedding efficiently via sparse matrix decomposition. Then we introduce a network propagation pattern via spectral analysis to incorporate local and global structure information into the embedding. Besides, this model can be generalized to integrate network information into other insufficiently trained embeddings at speed. Benefiting from sparse spectral network embedding, our experiment on four different datasets shows that Progle outperforms or is comparable to state-of-the-art unsupervised comparison approaches---DeepWalk, LINE, node2vec, GraRep, and HOPE, regarding accuracy, while is $10\times$ faster than the fastest word2vec-based method. Finally, we validate the scalability of Progle both in real large-scale networks and multiple scales of synthetic networks.
We consider the task of learning the parameters of a {\em single} component of a mixture model, for the case when we are given {\em side information} about that component, we call this the "search problem" in mixture models. We would like to solve this with computational and sample complexity lower than solving the overall original problem, where one learns parameters of all components. Our main contributions are the development of a simple but general model for the notion of side information, and a corresponding simple matrix-based algorithm for solving the search problem in this general setting. We then specialize this model and algorithm to four common scenarios: Gaussian mixture models, LDA topic models, subspace clustering, and mixed linear regression. For each one of these we show that if (and only if) the side information is informative, we obtain parameter estimates with greater accuracy, and also improved computation complexity than existing moment based mixture model algorithms (e.g. tensor methods). We also illustrate several natural ways one can obtain such side information, for specific problem instances. Our experiments on real data sets (NY Times, Yelp, BSDS500) further demonstrate the practicality of our algorithms showing significant improvement in runtime and accuracy.
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