Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Tensor completion using geodesics on Segre manifolds

Lars Swijsen, Joeri Van der Veken, Nick Vannieuwenhoven

TL;DR

This work develops a Riemannian conjugate gradient method for low-rank tensor completion by exploiting the geometry of the Segre manifold. A key contribution is the explicit unit-speed geodesic formula on Seg, which serves as an efficient retraction in the optimization over the product manifold Seg$^{\times r}$. The algorithm relies on the Riemannian gradient (projection of the ambient gradient), geodesic-based moves, and parallel transport, with a lightweight quadratic-interpolation line search. Empirically, the method achieves competitive tensor completion performance on incomplete data, demonstrated on MovieLens and fluorescence spectroscopy, and shows substantial data-efficiency, recovering meaningful decompositions from as little as about $9\%$ of the data.

Abstract

We propose a Riemannian conjugate gradient (CG) optimization method for finding low rank approximations of incomplete tensors. Our main contribution consists of an explicit expression of the geodesics on the Segre manifold. These are exploited in our algorithm to perform the retractions. We apply our method to movie rating predictions in a recommender system for the MovieLens dataset, and identification of pure fluorophores via fluorescent spectroscopy with missing data. In this last application, we recover the tensor decomposition from less than $10\%$ of the data.

Tensor completion using geodesics on Segre manifolds

TL;DR

This work develops a Riemannian conjugate gradient method for low-rank tensor completion by exploiting the geometry of the Segre manifold. A key contribution is the explicit unit-speed geodesic formula on Seg, which serves as an efficient retraction in the optimization over the product manifold Seg. The algorithm relies on the Riemannian gradient (projection of the ambient gradient), geodesic-based moves, and parallel transport, with a lightweight quadratic-interpolation line search. Empirically, the method achieves competitive tensor completion performance on incomplete data, demonstrated on MovieLens and fluorescence spectroscopy, and shows substantial data-efficiency, recovering meaningful decompositions from as little as about of the data.

Abstract

We propose a Riemannian conjugate gradient (CG) optimization method for finding low rank approximations of incomplete tensors. Our main contribution consists of an explicit expression of the geodesics on the Segre manifold. These are exploited in our algorithm to perform the retractions. We apply our method to movie rating predictions in a recommender system for the MovieLens dataset, and identification of pure fluorophores via fluorescent spectroscopy with missing data. In this last application, we recover the tensor decomposition from less than of the data.

Paper Structure

This paper contains 12 sections, 2 theorems, 43 equations, 6 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Introduction to Riemannian Geometry
  3. Geodesics on the Segre manifold
  4. Riemannian conjugate gradient algorithm for tensor decomposition
  5. Objective function
  6. Riemannian gradient
  7. Geodesics
  8. Parallel transport
  9. Line search
  10. Experimental results
  11. Recommender system for MovieLens data
  12. Fluorescence spectroscopy

Key Result

Theorem 1.1

The (unit-speed) geodesic of $Seg$ through the point in the direction is given by where $f(t)=\tan^{-1}\left(\frac{\sqrt{P ^2+1}}{\lambda}t+P\right)-\tan^{-1}(P)$, $P=\frac{\dot\lambda}{\lambda M}$ and $M=\sqrt{\sum_{k=1}^d||\dot x_k||^2}$.

Figures (6)

  • Figure 1: Error for different step-sizes
  • Figure 2: RMSE for different methods of predicting ratings.
  • Figure 3: Relative errors per rank
  • Figure 4: Relative error of the approximation obtained using only $\phi\cdot\dim(\sigma_r(Segre))$ elements.
  • Figure 5: Core consistency of the approximation obtained using only $\phi\cdot\dim(\sigma_r(Segre))$ elements.
  • ...and 1 more figures

Theorems & Definitions (3)

  • Theorem 1.1
  • Lemma 3.1
  • proof : Proof of \ref{['geo']}