Amin Nejatbakhsh

Gonzalo Mena, Amin Nejatbakhsh, Erdem Varol et al.

We study Sinkhorn EM (sEM), a variant of the expectation maximization (EM) algorithm for mixtures based on entropic optimal transport. sEM differs from the classic EM algorithm in the way responsibilities are computed during the expectation step: rather than assign data points to clusters independently, sEM uses optimal transport to compute responsibilities by incorporating prior information about mixing weights. Like EM, sEM has a natural interpretation as a coordinate ascent procedure, which iteratively constructs and optimizes a lower bound on the log-likelihood. However, we show theoretically and empirically that sEM has better behavior than EM: it possesses better global convergence guarantees and is less prone to getting stuck in bad local optima. We complement these findings with experiments on simulated data as well as in an inference task involving C. elegans neurons and show that sEM learns cell labels significantly better than other approaches.

Erdem Varol, Amin Nejatbakhsh, Conor McGrory

Motion segmentation for natural images commonly relies on dense optic flow to yield point trajectories which can be grouped into clusters through various means including spectral clustering or minimum cost multicuts. However, in biological imaging scenarios, such as fluorescence microscopy or calcium imaging, where the signal to noise ratio is compromised and intensity fluctuations occur, optical flow may be difficult to approximate. To this end, we propose an alternative paradigm for motion segmentation based on optimal transport which models the video frames as time-varying mass represented as histograms. Thus, we cast motion segmentation as a temporal non-linear matrix factorization problem with Wasserstein metric loss. The dictionary elements of this factorization yield segmentation of motion into coherent objects while the loading coefficients allow for time-varying intensity signal of the moving objects to be captured. We demonstrate the use of the proposed paradigm on a simulated multielectrode drift scenario, as well as calcium indicating fluorescence microscopy videos of the nematode Caenorhabditis elegans (C. elegans). The latter application has the added utility of extracting neural activity of the animal in freely conducted behavior.

Erdem Varol, Amin Nejatbakhsh

In this paper, we expand upon the theory of trend filtering by introducing the use of the Wasserstein metric as a means to control the amount of spatiotemporal variation in filtered time series data. While trend filtering utilizes regularization to produce signal estimates that are piecewise linear, in the case of $\ell_1$ regularization, or temporally smooth, in the case of $\ell_2$ regularization, it ignores the topology of the spatial distribution of signal. By incorporating the information about the underlying metric space of the pixel layout, the Wasserstein metric is an attractive choice as a regularizer to undercover spatiotemporal trends in time series data. We introduce a globally optimal algorithm for efficiently estimating the filtered signal under a Wasserstein finite differences operator. The efficacy of the proposed algorithm in preserving spatiotemporal trends in time series video is demonstrated in both simulated and fluorescent microscopy videos of the nematode caenorhabditis elegans and compared against standard trend filtering algorithms.

Amin Nejatbakhsh, Erdem Varol

We propose methods for estimating correspondence between two point sets under the presence of outliers in both the source and target sets. The proposed algorithms expand upon the theory of the regression without correspondence problem to estimate transformation coefficients using unordered multisets of covariates and responses. Previous theoretical analysis of the problem has been done in a setting where the responses are a complete permutation of the regressed covariates. This paper expands the problem setting by analyzing the cases where only a subset of the responses is a permutation of the regressed covariates in addition to some covariates being outliers. We term this problem \textit{robust regression without correspondence} and provide several algorithms based on random sample consensus for exact and approximate recovery in a noiseless and noisy one-dimensional setting as well as an approximation algorithm for multiple dimensions. The theoretical guarantees of the algorithms are verified in simulated data. We demonstrate an important computational neuroscience application of the proposed framework by demonstrating its effectiveness in a \textit{Caenorhabditis elegans} neuron matching problem where the presence of outliers in both the source and target nematodes is a natural tendency.