On Manifold Learning in Plato's Cave: Remarks on Manifold Learning and Physical Phenomena

Many techniques in machine learning attempt explicitly or implicitly to infer a low-dimensional manifold structure of an underlying physical phenomenon from measurements without an explicit model of the phenomenon or the measurement apparatus. This paper presents a cautionary tale regarding the discrepancy between the geometry of measurements and the geometry of the underlying phenomenon in a benign setting. The deformation in the metric illustrated in this paper is mathematically straightforward and unavoidable in the general case, and it is only one of several similar effects. While this is not always problematic, we provide an example of an arguably standard and harmless data processing procedure where this effect leads to an incorrect answer to a seemingly simple question. Although we focus on manifold learning, these issues apply broadly to dimensionality reduction and unsupervised learning.

Image reconstruction in light-sheet microscopy: spatially varying deconvolution and mixed noise

We study the problem of deconvolution for light-sheet microscopy, where the data is corrupted by spatially varying blur and a combination of Poisson and Gaussian noise. The spatial variation of the point spread function (PSF) of a light-sheet microscope is determined by the interaction between the excitation sheet and the detection objective PSF. First, we introduce a model of the image formation process that incorporates this interaction, therefore capturing the main characteristics of this imaging modality. Then, we formulate a variational model that accounts for the combination of Poisson and Gaussian noise through a data fidelity term consisting of the infimal convolution of the single noise fidelities, first introduced in L. Calatroni et al. "Infimal convolution of data discrepancies for mixed noise removal", SIAM Journal on Imaging Sciences 10.3 (2017), 1196-1233. We establish convergence rates in a Bregman distance under a source condition for the infimal convolution fidelity and a discrepancy principle for choosing the value of the regularisation parameter. The inverse problem is solved by applying the primal-dual hybrid gradient (PDHG) algorithm in a novel way. Finally, numerical experiments performed on both simulated and real data show superior reconstruction results in comparison with other methods.