| Description | TThe quantification of entanglement via linear semi-infinite programs has recently received attention Although the technique established in this paper provides an approximation of the value of entanglement, one can do better: a recent algorithm in robust optimization permits us to numerically compute the precise value. This project is about employing this algorithm, adapting it slightly if needed, and arriving at the precise value of entanglement. |
|---|---|
| Number of students | 1 |
| Year of study | Students in their 3rd year (Semester 5), Students in their 4th/5th year (Semester 7/9) |
| CPI | 8.5 and above |
| Prerequisites | Standard background on probability and optimization; the rest will be built. |
| Duration | 6 months to 1 year |
| Learning outcome | Learning a novel technique in robust optimization and its application to solve a problem in quantum physics |
| Weekly time commitment | 4 hours |
| General expectations | Serious Students Only |
| Assignment | https://doi.org/10.1103/PhysRevA.104.022413 and https://doi.org/10.1007/s10479-022-04810-4 |
| Instructions for assignment | Study the two papers and summarize them |
| Description | "This project centers around a novel approach to learning from data a la function approximation, and the Chebyshev center problem is the centerpiece in this context. Given a compact subset K of R^n, the Chebyshev center problem consists of finding a minimal circumscribing ball (a so-called Chebyshev ball) containing K. In the context of learning from data, the problem translates to finding the center of a Chebyshev ball described by a dictionary of functions justifying the data. Finding the Chebyshev ball is an NP hard problem in general, but recent advances have established ideas for numerically tractable sharp approximations and the non-noisy case (i.e., optimal interpolation) has been worked out in detail. This project is about carrying out the computations in the case of noisy data. Efforts will be split equally between learning new theory and developing numerical tools for the aforementioned applications, and there is a strong possibility of writing journal papers and filing patents." |
|---|---|
| Number of students | 2 |
| Year of study | Students entering 3rd year, Students entering 4th/5th year |
| CPI | 8.5 and above |
| Prerequisites | Background in optimization and probability |
| Duration | 3 months extendable up to 3 years |
| Learning outcome | Introduction to a novel numerical apparatus and the challenging problem of optimal learning from data |
| Weekly time commitment | At least 20 during the vacation, at least 5 during the semester |
| General expectations | Sincerity |
| Assignment |
https://doi.org/10.48550/arXiv.2203.15994 https://doi.org/10.1007/s10479-022-04810-4 |
| Instructions for assignment | The first document contains a discussion of the Chebyshev center problem in the context of optimal learning (non-noisy case). Their assertion concerning the quality of approximation possible (equation (3.11)) is suboptimal, and the factor C there can be made equal to 1! The germ of this idea is in the second document. Study the first carefully to get an idea of the math involved. The numerical algorithm has to be designed along the lines of the second document. |
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