Log-Hilbert-Schmidt metric between positive definite operators on Hilbert spaces
This paper introduces a novel mathematical and computational framework, namely Log-Hilbert-Schmidt metric between positive definite operators on a Hilbert space. This is a generalization of the Log-Euclidean metric on the Riemannian manifold of positive definite matrices to the infinite-dimensional setting. The general framework is applied in particular to compute distances between covariance operators on a Reproducing Kernel Hilbert Space (RKHS), for which we obtain explicit formulas via the corresponding Gram matrices. Empirically, we apply our formulation to the task of multi-category image classification, where each image is represented by an infinite-dimensional RKHS covariance operator. On several challenging datasets, our method significantly outperforms approaches based on covariance matrices computed directly on the original input features, including those using the Log-Euclidean metric, Stein and Jeffreys divergences, achieving new state of the art results.
Summary: This paper extends the idea of Log-Euclidean metric (LEM) on the Riemannian manifold of positive definite matrices to an infinite-dimensional setting. In this work, the proposed Log-Hilbert-Schmidt metric (LHSM) provides ways of computing distance between positive definite operators on a Hilbert space. From a theoretical perspective, extending the LEM to an infinite dimensional space is not a trivial task because one cannot obtain the infinite-dimensional counterpart of LEM by simply letting the dimension approach infinity. There are two fundamental problems which have been addressed in this paper. First, if A is a self-adjoint operator in an infinite-dimensional Hilbert space, log(A), which is an important ingredience in Log-Euclidean metric, is not well-defined and may not be bounded. Second, the identity operator in this space is not Hilbert-Schmidt and would have infinite distance from any Hilbert-Schmidt operator. This paper gives a rigorous solutions to the aforementioned problems. Moreover, when the covariance operator is defined on an RKHS induced by a positive definite kernel, the paper gives an explicit formulas for the proposed metric that can be evaluated entirely via the corresponding Gram matrix.From application point of view, the proposed metric is applied in the task of multi-category image classification in which each image is represented by an infinite-dimensional RKHS covariance operator constructed from "original features" in the input space. The proposed metric is found to outperform the state-of-the-art algorithm which only considers the finite covariance matrix in the input space. However, it will be more interesting to compare this method to other image classification approaches, e.g., those based on deep learning.The idea presented in this paper makes sense intuitively, and will be very useful in practice because one can view this framework as a non-linear extension of learning problems when the samples are represented by the covariance matrices. In addition to images, I believe one can broadly apply this metric to other kinds of data which have high-order correlation. Moreover, defining the metric in the RKHS allows even more flexibility as one can freely choose different kernel functions.Quality:The paper is of high quality. The detail of the claims and proofs are given in great detail.Clarity:The paper is clearly written and I am able to follow the logic.Originality:The novelty of this work is the rigorous formulation of infinite-version of Log-Euclidean metric. The problem is not trivial and I think the contribution of this work in term of technical results is quite substantial.Significance:The proposed metric is shown to outperform state-of-the-art algorithms, so the contribution of this work should be significant in practice. However, the authors didn't compare different approaches in term of computational complexity, so it is difficult to conclude from this aspect.Some minor comments:- If I understand correctly, the covariance operator is defined for each sample, and therefore is fundamentally different from the standard way that the covariance operator has been defined such as in kernel PCA. That is, the covariance operator is constructed from a sample and as a sample size goes to infinity, we obtain the population counterpart of the underlying distribution. On the other hand, in this work each image is represented by a covariance operator. Do you have an intuitive explanation of what the population counterpart of this covariance operator is? Are they fundamentally the same or different?- In section 6, what are the "original features" extracted from the image and how exactly these features are generated?