https://us02web.zoom.us/j/86785731080?pwd=dk10aVJNM3NXeHE3WFJvZmlUQndadz09
Meeting ID: 867 8573 1080
Passcode: TAUEESYS 

Speaker: Dr.  Nadav Dym

Department of Mathematics of Duke University

Monday, November 9^th, 2020, at 15:00

On the injectivity and (in)stability of invariant encoding

Abstract

            Quotient spaces are a natural mathematical tool to describe a variety of algorithmic problems in computer vision and related fields, where different objects are to be compared while their natural symmetries are to be ignored.

 One popular approach for working in quotient spaces is cancelling out symmetries  by finding a  canonical alignment of the objects at hand. This approach typically involves solving a challenging optimization problem over the symmetry group. We will review the tractability of these problems for sets, their intractability for graphs, and recent advances in the complexity of computing these problems for the intermediate case of rigid point clouds.

An alternative approach for working in quotient spaces avoids the computational challenges of alignment by directly mapping objects to features invariant to the object’s symmetries. The challenge with this approach is devising a sufficiently expressive family of invariant features. We will describe constructive methods for mapping quotient spaces injectively to a finite set of invariants, which lead to proofs of universality of networks for sets and rigid point clouds, and to injectivity of linear phaseless measurements in the context of phase retrieval problems. We will then discuss results in the phase retrieval literature that suggest that though injective, mappings to invariant features can be unstable and suffer from the curse of dimensionality. These results indicate that every system of invariants has `blind spots’, which corresponds to `disconnected signals’ in the measurement domain. We will describe work to quantify the stability via measures of connectivity used in spectral graph theory, and in particular show that real phase retrieval is related to the Cheeger constant, while complex phase retrieval relates to the second eigenvalue of the Laplacian.

Short Bio

I am an applied mathematician and computer scientist, interested in the development and analysis of algorithms, typically targeting 3D problems in computer vision and related fields. I am currently working on problems related to global optimization, phase retrieval, and theoretical and geometric deep learning. I am an Assistant Research Professor at the Department of Mathematics of Duke University, hosted by Prof. Ingrid Daubechies at the Information Initiative at Duke (iiD). In 2018 I completed my PhD in the Department of Computer Science and Applied Mathematics at the Weizmann Institute of Science under the supervision of Prof. Yaron Lipman. I have received a BSc and MSc in mathematics from the Hebrew University.

השתתפות בסמינר תיתן קרדיט שמיעה = עפ"י רישום שם מלא + מספר ת.ז.  בצ'אט

Dear All,
 

You are cordially invited to the LMI seminar, that will be broadcasted on, Nov 4, 2020 01:00 PM Jerusalem

Abstract:

Color is a fundamental contrast mechanism in fluorescence microscopy, providing the basis for numerous imaging and spectroscopy techniques. The ever-growing need to acquire high-throughput, dynamic data from multicolor species is driving the development of optical schemes that optimize the achievable spectral, temporal, and spatial resolution needed in order to follow biological, chemical and physical processes. Here we introduce Continuously Controlled Spectral-resolution (CoCoS) microscopy, an imaging scheme that encodes color into spatial read-out in the image plane, with continuous control over the spectral resolution. The concept enables single-frame acquisition of multiple color channels, allowing simultaneous, single-molecule colocalization for barcoding and Förster resonance energy transfer (FRET) experiments. The simple control over the spectral dispersion allows switching between imaging modalities at a click of a button. We demonstrate the utility of CoCoS for multicolor localization microscopy of microRNA barcodes in clinical samples, single-molecule FRET measurements, and single-molecule spectroscopy. CoCoS may be integrated as a simple add-on to existing microscopes and will find use in applications that aim to record dynamic, multicolor localization events such as in multiplex FRET and tracking of multi-component, interacting complexes.