Research summary

I am currently a postdoc in the Imaging and Computing Group with Prof. Laurent Demanet at the Department of Mathematics, MIT.

My main research area is broadly in applied and computational mathematics, and more specifically, applied harmonic analysis, numerical analysis and inverse problems. I develop mathematical models which are both useful in applications, and computationally stable vis. mismatch and uncertainty; key research questions are quantifying the best possible trade-off between stability and complexity, and developing optimal recovery algorithms.

Between 2015-2016  I was a postdoc with Prof. Miki Elad at the Technion, Israel. I obtained my Ph.D. from the Weizmann Institute, Israel in 2014, under the supervision of Prof. Yosef Yomdin.

Stable exponential fitting and sparse super-resolution

Super-resolution is an ill-posed inverse problem with great variety of ramifications. A basic task is to recover few close-by “point-like” sources (scatterers in remote sensing, target DOAs, spectral lines in NMR, neural spike timings, narrowband sources in wireless communications) from low- pass and inaccurate data, and it is important to understand the fundamental limits of this highly nonlinear problem. In collaboration with multiple researchers, I have derived the best possible resolution limits depending on the signal’s local and global complexity, data bandwidth, and the noise level. The stability theory provides sharp bounds for the super- resolution factor for each source, depending on how close is this source to other sources. I have also developed optimal algorithms achieving these bounds in certain cases.

For further info:

1. Batenkov, D. & Yomdin, Y. On the accuracy of solving confluent Prony systems. SIAM J. Appl. Math.73,134–154 (2013).
2. Akinshin, A., Batenkov, D. & Yomdin, Y. Accuracy of spike-train Fourier reconstruction for colliding nodes. in 2015 International Conference on Sampling Theory and Applications (SampTA)617–621 (2015). doi:10.1109/SAMPTA.2015.7148965
3. Batenkov, D. Accurate solution of near-colliding Prony systems via decimation and homotopy continuation. Theoretical Computer Science681,27–40 (2017).
4. Batenkov, D. Stability and super-resolution of generalized spike recovery. Applied and Computational Harmonic Analysis45,299–323 (2018).
5. Batenkov, D., Demanet, L., Goldman, G. & Yomdin, Y. Stability of partial Fourier matrices with clustered nodes. arXiv:1809.00658 [cs, math](2018).
6. Batenkov, D., Demanet, L. & Mhaskar, H. N. Stable soft extrapolation of entire functions. Inverse Problems, accepted for publicationarXiv:1806.09749 [math](2018).

Resolution of the Gibbs phenomenon

Accurate reconstruction of a piecewise-smooth function from bandlimited data is a challenging approximation problem. The Gibbs artifacts near the jump discontinuities severely reduce the convergence rate of the Fourier partial sum, with repercussions in applications such as medical imaging, shock capturing in nonlinear flows, and data compression. I have developed a provably order-optimal nonlinear approximation algorithm for this problem, providing reconstruction accuracy (both for the jump locations, magnitudes and the function values between the jumps) which is commensurate with the smoothness of the pieces. These results and methods open several venues for future research, including development of efficient numerical solvers, extension to higher-dimensionsal settings (e.g. distributions on compact Riemannian manifolds, with potential applications in astrophysics, geodesy, source localization and machine learning), applications to computed tomography/MRI (wavefront set reconstruction), change-point detection in statistics.

For further info:

1. Batenkov, D. & Yomdin, Y. Algebraic Fourier reconstruction of piecewise smooth functions. Mathematics of Computation 81,277–318 (2012).
2. Batenkov, D., Friedland, O. & Yomdin, Y. Sampling, Metric Entropy, and Dimensionality Reduction. SIAM J. Math. Anal. 47,786–796 (2015).
3. Batenkov, D. Complete algebraic reconstruction of piecewise-smooth functions from Fourier data. Math. Comp. 84,2329–2350 (2015).

From global to local priors

For very complex and high-dimensional data such as images, global modeling of the signal is infeasible. Instead, it has been shown experimentally that small pieces of a signal can be sparsely represented in an appropriate dictionary of atoms. A general open problem is to infer useful properties of the global signal, implicitly defined by the local sparsity assumptions. We have initiated a systematic investigation of such “bottom-up” priors, and we have found that indeed, the resulting local-global connection can be effectively used for better recovery. There is much more to be done, including designing efficient denoising algorithms, and learning local dictionaries from data.

For further info:

1. Batenkov, D., Romano, Y. & Elad, M. On the Global-Local Dichotomy in Sparsity Modeling. in Compressed Sensing and its Applications 1–53 (Birkhäuser, Cham, 2017). doi:10.1007/978-3-319-69802-1_1