In many modern large-scale learning applications, the amount of unlabeled data far exceeds that of labeled data. A common instance of this problem is the transductive setting where the unlabeled test points are known to the learning algorithm. This paper presents a study of regression problems in that setting. It presents explicit VC-dimension error bounds for transductive regression that hold for all bounded loss functions and coincide with the tight classification bounds of Vapnik when applied to classification. It also presents a new transductive regression algorithm inspired by our bound that admits a primal and kernelized closedform solution and deals efficiently with large amounts of unlabeled data. The algorithm exploits the position of unlabeled points to locally estimate their labels and then uses a global optimization to ensure robust predictions. Our study also includes the results of experiments with several publicly available regression data sets with up to 20,000 unlabeled examples. The comparison with other transductive regression algorithms shows that it performs well and that it can scale to large data sets.

Applications of our proposed technique include embedding with a natural out-of-sample extension and tasks such as tangent distance estimation, frame rate up-conversion, video compression and motion transfer.

Neural motor prostheses (NMPs) require the accurate decoding of motor cortical population activity for the control of an artificial motor system. Previous work on cortical decoding for NMPs has focused on the recovery of hand kinematics. Human NMPs however may require the control of computer cursors or robotic devices with very different physical and dynamical properties. Here we show that the firing rates of cells in the primary motor cortex of nonhuman primates can be used to control the parameters of an artificial physical system exhibiting realistic dynamics. The model represents 2D hand motion in terms of a point mass connected to a system of idealized springs. The nonlinear spring coefficients are estimated from the firing rates of neurons in the motor cortex. We evaluate linear and a nonlinear decoding algorithms using neural recordings from two monkeys performing two different tasks. We found that the decoded spring coefficients produced accurate hand trajectories compared with state-of-the-art methods for direct decoding of hand kinematics. Furthermore, using a physically-based system produced decoded movements that were more "natural" in that their frequency spectrum more closely matched that of natural hand movements.

Semi-supervised learning is more powerful than supervised learning by using both labeled and unlabeled data. In particular, the manifold regularization framework, together with kernel methods, leads to the Laplacian SVM (LapSVM) that has demonstrated state-of-the-art performance. However, the LapSVM solution typically involves kernel expansions of all the labeled and unlabeled examples, and is slow on testing. Moreover, existing semi-supervised learning methods, including the LapSVM, can only handle a small number of unlabeled examples. In this paper, we integrate manifold regularization with the core vector machine, which has been used for large-scale supervised and unsupervised learning. By using a sparsified manifold regularizer and formulating as a center-constrained minimum enclosing ball problem, the proposed method produces sparse solutions with low time and space complexities. Experimental results show that it is much faster than the LapSVM, and can handle a million unlabeled examples on a standard PC; while the LapSVM can only handle several thousand patterns.

We describe Hidden Semi-Markov Support Vector Machines (SHM SVMs), an extension of HM SVMs to semi-Markov chains. This allows us to predict segmentations of sequences based on segment-based features measuring properties such as the length of the segment. We propose a novel technique to partition the problem into sub-problems. The independently obtained partial solutions can then be recombined in an efficient way, which allows us to solve label sequence learning problems with several thousands of labeled sequences. We have tested our algorithm for predicting gene structures, an important problem in computational biology. Results on a well-known model organism illustrate the great potential of SHM SVMs in computational biology.

We derive a cost functional for estimating the relationship between highdimensional observations and the low-dimensional process that generated them with no input-output examples. Limiting our search to invertible observation functions confers numerous benefits, including a compact representation and no suboptimal local minima. Our approximation algorithms for optimizing this cost functional are fast and give diagnostic bounds on the quality of their solution. Our method can be viewed as a manifold learning algorithm that utilizes a prior on the low-dimensional manifold coordinates. The benefits of taking advantage of such priors in manifold learning and searching for the inverse observation functions in system identification are demonstrated empirically by learning to track moving targets from raw measurements in a sensor network setting and in an RFID tracking experiment.

In many areas of science and engineering, the problem arises how to discover low dimensional representations of high dimensional data. Recently, a number of researchers have converged on common solutions to this problem using methods from convex optimization. In particular, many results have been obtained by constructing semidefinite programs (SDPs) with low rank solutions. While the rank of matrix variables in SDPs cannot be directly constrained, it has been observed that low rank solutions emerge naturally by computing high variance or maximal trace solutions that respect local distance constraints. In this paper, we show how to solve very large problems of this type by a matrix factorization that leads to much smaller SDPs than those previously studied. The matrix factorization is derived by expanding the solution of the original problem in terms of the bottom eigenvectors of a graph Laplacian. The smaller SDPs obtained from this matrix factorization yield very good approximations to solutions of the original problem. Moreover, these approximations can be further refined by conjugate gradient descent. We illustrate the approach on localization in large scale sensor networks, where optimizations involving tens of thousands of nodes can be solved in just a few minutes.

Large numbers of descriptors and large codebooks are needed for good results and this becomes slow using k-means. We introduce Extremely Randomized Clustering Forests - ensembles of randomly created clustering trees - and show that these provide more accurate results, much faster training and testing and good resistance to background clutter in several state-of-the-art image classification tasks.

We describe a novel unsupervised method for learning sparse, overcomplete features. The model uses a linear encoder, and a linear decoder preceded by a sparsifying non-linearity that turns a code vector into a quasi-binary sparse code vector. Given an input, the optimal code minimizes the distance between the output of the decoder and the input patch while being as similar as possible to the encoder output. Learning proceeds in a two-phase EMlike fashion: (1) compute the minimum-energy code vector, (2) adjust the parameters of the encoder and decoder so as to decrease the energy. The model produces "stroke detectors" when trained on handwritten numerals, and Gabor-like filters when trained on natural image patches. Inference and learning are very fast, requiring no preprocessing, and no expensive sampling. Using the proposed unsupervised method to initialize the first layer of a convolutional network, we achieved an error rate slightly lower than the best reported result on the MNIST dataset. Finally, an extension of the method is described to learn topographical filter maps.

We use multi-electrode recordings from cat primary visual cortex and investigate whether a simple linear classifier can extract information about the presented stimuli. We find that information is extractable and that it even lasts for several hundred milliseconds after the stimulus has been removed. In a fast sequence of stimulus presentation, information about both new and old stimuli is present simultaneously and nonlinear relations between these stimuli can be extracted. These results suggest nonlinear properties of cortical representations. The important implications of these properties for the nonlinear brain theory are discussed.