Abstract: In this talk, I will review some recent advances on deep generative models through the prism of Optimal Transport (OT). OT provides a way to define robust loss functions to perform high dimensional density fitting using generative models. This defines so called Minimum Kantorovitch Estimators (MKE) [1]. This approach is especially useful to recast several unsupervised deep learning methods in a unifying framework. Most notably, as shown respectively in [2,3] (and reviewed in [4]) Variational Autoencoders (VAE) and Generative Adversarial Networks (GAN) can be interpreted as (respectively primal and and dual) approximate MKE. This is a joint work with Aude Genevay and Marco Cuturi.
Abstract: Quantized random embeddings are an efficient dimensionality reduction technique which preserves the distances of low-complexity signals up to some controllable additive and multiplicative distortions. In this talk, we instead focus on verifying when this technique preserves the separability of two disjoint closed convex sets, i.e., in a quantized view of the "rare eclipse problem" introduced by Bandeira et al. in 2014. This separability would ensure exact classification of signals in such sets from the signatures output by this non-linear dimensionality reduction. We here present a result relating the embedding's dimension, its quantizer resolution and the sets' separation, as well as some numerically testable conditions to illustrate it. Experimental evidence is then provided in the special case of two ℓ2-balls, tracing the phase transition curves that ensure these sets' separability in the embedded domain. This is a joint work with Valerio Cambareri and Chunlei Xu.
Abstract: Compressive Sensing enables to reconstruct sparse signals from a limited number of linear measurements. Thanks to the restricted isometry property (RIP), some decoding algorithms, such as the iterative hard thresholding (IHT) algorithm, are theoretically guaranteed to find a best K-term approximation of an unknown signal. Whereas, in practice, due to the digitalisation, we often observe the nonlinear quantised measurements, taking a discrete value in a finite set for each measurement. Thus, the magnitude information of a signal is lost due to such a lossy quantisation. Similarly to the IHT algorithm, the quantised iterative hard thresholding (QIHT) algorithms is proposed to possibly tackle the quantised compressive sensing (QCS) problem. Now, the challenge is how to achieve a small reconstruction error by solving the QCS problem via the QIHT algorithm.
In this talk, we consider the dithered uniform scaler quantisation, which is adding a random uniform dithering vector to a random linear map and applying the uniform scaler quantisation operator afterwards. Thanks to this random dithering vector, we can prove that the first step of the QIHT can already provide a relatively good k-term estimation of an unknown signal when the sensing matrix satisfies the so-called quantised inner product embedding (QIPE) property. Furthermore, if the unknown signal is not sparse, but belongs to a more general set, such as the union of subspaces, a convex set or even a set of low rank matrix, we can show that the projected back projection of the quantised measurements onto these general sets also can provide a relatively good estimation of the unknown signal under certain QIPE conditions. Lastly, we conduct some numerical tests to verify theoretical results given by the QIPE conditions.
Abstract: Multiple scattering of an electromagnetic wave as it passes through an object is a fundamental problem that limits the performance of current imaging systems. From the perspective of imaging inverse problems, multiple scattering leads to nonlinear forward models that generally lead to intractable optimization problems. In this talk, I will discuss recent advances for designing optimization schemes that can account for multiple scattering while also accommodating model-based priors for imaging.
Abstract: Computational imaging methods that can exploit multiple modalities have the potential to enhance the capabilities of traditional sensing systems. In this work, we propose a new method that reconstructs multimodal images from their linear measurements by exploiting redundancies across different modalities.
Our method combines a convolutional group-sparse representation of images with Total Variation regularization for high-quality multimodal imaging. We develop an online algorithm that enables the unsupervised learning of convolutional dictionaries on large-scale datasets that are typical in such applications. We illustrate the benefit of our approach in the context of joint intensity-depth imaging. The work has been accepted at ICIP2017.
Abstract: Unmanned aerial vehicle control is a hot topic in research and at the crossroad of a lot of disciplines. For instance, safe and reliable navigation of UAVs requires consideration of the surrounding environment, in particular, the external wind conditions. This external wind configuration is usually evaluated by computationnaly expensive efforts and depends on external meteorological parameters. Hopefully, it can be modelled as a Gaussian process characterized by a covariance matrix belonging to the space of PSD matrices of rank r. In this work, we both expoit the manifold structure of this specific space and also propose a method to fit a small set of pre-computed solutions. That way, for a new value of the external meteorological parameters, we are able to recover a sufficiently accurate wind field configuration in a computationnaly tractable effort. Our method is based on manifold-valued Bezier curves. Joint work with E. M. Massart, A. Musolas, P.-A. Absil, J.M. Hendrickx, L. Jacques and Y. Marzouk.