Geometric Structures seminar - Yosef Yomdin - Tuesday June 11th, 2024 , 14:00 (Rome time) - room 005 at SISSA

[Samuel Borza]

Dear All,

This is to announce the next seminar from the series "Geometric Structures".

Speaker: Yosef Yomdin (Weizmann Institute of Science)

Title: Super-resolution, classical Moment Theory, and some Real Algebraic Geometry
Time: Tuesday June 11th, 2024 , 14:00 (Rome time)
Venue: room 005 - SISSA main building

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Abstract:
We consider the problem of reconstruction of “spike-train” signals

(1)       F(x) = \sum_{j=1}^d  a_j \delta(x-x_j),

which are linear combinations of shifted delta-functions, from noisy Moment measurements

(2)       m_k(F) = \int F(x) x^k dx = \sum_{j=1}^d  a_j x_j^k.

This is equivalent to solving the so – called Prony system of algebraic equations

(3)       \sum_{j=1}^d  a_i  x_j^k = m_k(F),       k = 0,1,…, 2d-1,

with respect to the unknowns  (a_j, x_j),  j = 1,…,d.

Our goal is to understand the “intrinsic geometry” of the error amplification in the reconstruction process, stressing the case where the nodes x_j nearly collide. We study the geometry of system (3), independently of a specific reconstruction algorithm.

We construct a growing chain of algebraic sub-varieties Y_q (which we call Prony varieties) in the space of the parameters (a_j, x_j), which accurately control the rate of the error amplification. These sub-varieties Y_q can be reconstructed from the noisy moment measurements with a significantly better accuracy than the amplitudes a_j and the nodes x_j themselves. This opens a possibility to apply adaptive reconstruction algorithms, subordinated to the chain of the Prony varieties Y_q. We show that this approach in many cases provides higher reconstruction accuracy than the standard ones.
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More information can be found here: https://sites.google.com/view/geometric-structures/

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Samuël Borza