Out-of-sample generalizations for supervised manifold learning for classification

Supervised manifold learning methods for data classification map data samples residing in a high-dimensional ambient space to a lower-dimensional domain in a structure-preserving way, while enhancing the separation between different classes in the learned embedding. Most nonlinear supervised manifold learning methods compute the embedding of the manifolds only at the initially available training points, while the generalization of the embedding to novel points, known as the out-of-sample extension problem in manifold learning, becomes especially important in classification applications. In this work, we propose a semi-supervised method for building an interpolation function that provides an out-of-sample extension for general supervised manifold learning algorithms studied in the context of classification. The proposed algorithm computes a radial basis function (RBF) interpolator that minimizes an objective function consisting of the total embedding error of unlabeled test samples, defined as their distance to the embeddings of the manifolds of their own class, as well as a regularization term that controls the smoothness of the interpolation function in a direction-dependent way. The class labels of test data and the interpolation function parameters are estimated jointly with a progressive procedure. Experimental results on face and object images demonstrate the potential of the proposed out-of-sample extension algorithm for the classification of manifold-modeled data sets

Improved Heterogeneous Distance Functions

Instance-based learning techniques typically handle continuous and linear input values well, but often do not handle nominal input attributes appropriately. The Value Difference Metric (VDM) was designed to find reasonable distance values between nominal attribute values, but it largely ignores continuous attributes, requiring discretization to map continuous values into nominal values. This paper proposes three new heterogeneous distance functions, called the Heterogeneous Value Difference Metric (HVDM), the Interpolated Value Difference Metric (IVDM), and the Windowed Value Difference Metric (WVDM). These new distance functions are designed to handle applications with nominal attributes, continuous attributes, or both. In experiments on 48 applications the new distance metrics achieve higher classification accuracy on average than three previous distance functions on those datasets that have both nominal and continuous attributes.Comment: See http://www.jair.org/ for an online appendix and other files accompanying this articl

DeepVoxels: Learning Persistent 3D Feature Embeddings

In this work, we address the lack of 3D understanding of generative neural networks by introducing a persistent 3D feature embedding for view synthesis. To this end, we propose DeepVoxels, a learned representation that encodes the view-dependent appearance of a 3D scene without having to explicitly model its geometry. At its core, our approach is based on a Cartesian 3D grid of persistent embedded features that learn to make use of the underlying 3D scene structure. Our approach combines insights from 3D geometric computer vision with recent advances in learning image-to-image mappings based on adversarial loss functions. DeepVoxels is supervised, without requiring a 3D reconstruction of the scene, using a 2D re-rendering loss and enforces perspective and multi-view geometry in a principled manner. We apply our persistent 3D scene representation to the problem of novel view synthesis demonstrating high-quality results for a variety of challenging scenes.Comment: Video: https://www.youtube.com/watch?v=HM_WsZhoGXw Supplemental material: https://drive.google.com/file/d/1BnZRyNcVUty6-LxAstN83H79ktUq8Cjp/view?usp=sharing Code: https://github.com/vsitzmann/deepvoxels Project page: https://vsitzmann.github.io/deepvoxels

Asymptotic learning curves of kernel methods: empirical data v.s. Teacher-Student paradigm

How many training data are needed to learn a supervised task? It is often observed that the generalization error decreases as $n^{-\beta}$ where $n$ is the number of training examples and $\beta$ an exponent that depends on both data and algorithm. In this work we measure $\beta$ when applying kernel methods to real datasets. For MNIST we find $\beta\approx 0.4$ and for CIFAR10 $\beta\approx 0.1$, for both regression and classification tasks, and for Gaussian or Laplace kernels. To rationalize the existence of non-trivial exponents that can be independent of the specific kernel used, we study the Teacher-Student framework for kernels. In this scheme, a Teacher generates data according to a Gaussian random field, and a Student learns them via kernel regression. With a simplifying assumption -- namely that the data are sampled from a regular lattice -- we derive analytically $\beta$ for translation invariant kernels, using previous results from the kriging literature. Provided that the Student is not too sensitive to high frequencies, $\beta$ depends only on the smoothness and dimension of the training data. We confirm numerically that these predictions hold when the training points are sampled at random on a hypersphere. Overall, the test error is found to be controlled by the magnitude of the projection of the true function on the kernel eigenvectors whose rank is larger than $n$. Using this idea we predict relate the exponent $\beta$ to an exponent $a$ describing how the coefficients of the true function in the eigenbasis of the kernel decay with rank. We extract $a$ from real data by performing kernel PCA, leading to $\beta\approx0.36$ for MNIST and $\beta\approx0.07$ for CIFAR10, in good agreement with observations. We argue that these rather large exponents are possible due to the small effective dimension of the data.Comment: We added (i) the prediction of the exponent $\beta$ for real data using kernel PCA; (ii) the generalization of our results to non-Gaussian data from reference [11] (Bordelon et al., "Spectrum Dependent Learning Curves in Kernel Regression and Wide Neural Networks"