Section: New Results

High-Dimensional and Deep Regression

One of the most important achievements for the last years has been the development of high-dimensional to low-dimensional regression methods. The motivation for investigating this problem raised from several problems that appeared both in audio signal processing and in computer vision. Indeed, often the task in data-driven methods is to recover low-dimensional properties and associated parameterizations from high-dimensional observations. Traditionally, this can be formulated as either an unsupervised method (dimensionality reduction of manifold learning) or a supervised method (regression). We developed a learning methodology at the crossroads of these two alternatives: the output variable can be either fully observed or partially observed. This was cast into the framework of linear-Gaussian mixture models in conjunction with the concept of inverse regression. It gave rise to several closed-form and approximate inference algorithms [8]. The method is referred to as Gaussian locally linear mapping, or GLLiM. As already mentioned, high-dimensional regression is useful in a number of data processing tasks because the sensory data often lies in high-dimensional spaces. Each one of these tasks required a special-purpose version of our general framework. Sound-source localization was the first to benefit from our formulation. Nevertheless, the sparse nature of speech spectrograms required the development of a GLLiM version that is able to with full-spectrum sounds and to test with sparse-spectrum ones [9]. This could be immediately applied to audio-visual alignment and to sound-source separation and localization [7].

In conjunction with our computer vision work, high-dimensional regression is a very useful methodology since visual features, obtained either by hand-crafted feature extraction methods or using convolutional neural networks, lie in high-dimensional spaces. Such properties as object pose lie in low-dimensional spaces and must be extracted from features. We took such an approach and proposed a head pose estimator [10]. Visual tracking can also benefit from GLLiM. Indeed, it is not practical to track objects based on high-dimensional features. We therefore combined GLLiM with switching linear dynamic systems. In 2018 we proposed a robust deep regression method [46]. In parallel we thoroughly benchmarked and analyzed deep regression tasks using several CNN architectures [57].