IMI Interdisciplinary Mathematics InstituteCollege of Arts and Sciences

 Carolina Distinguished Professor Department of Mathematics University of South Carolina Office: LC 317P Phone: 803-777-6456 Email: temlyak@math.sc.edu

## Education

 Habilitation Mathematics Steklov Institute of Mathematics, Russia 1981 Ph.D. Mathematics Steklov Institute of Mathematics, Russia 1978 M.S. Mathematics Moscow Institute of Physics and Technology, Russia 1976

## Experience

 2007 – Present Carolina Distinguished Professor Department of Mathematics, University of South Carolina 1992 – Present Associate Member Steklov Institute of Mathematics, Russia 1995 – 2007 Professor Department of Mathematics, University of South Carolina 1992 – 1995 Associate Professor Department of Mathematics, University of South Carolina 1986 – 1992 Leading Scientist Steklov Institute of Mathematics, Russia 1986 – 1992 Professor (part-time) Moscow Institute of Physics and Technology, Russia 1979 – 1986 Assistant/Associate Professor (part-time) Moscow Institute of Physics and Technology, Russia 1978 – 1986 Scientist Steklov Institute of Mathematics, Russia

## Research

### Research Interests

• Greedy approximation. I continue to actively work in both greedy approximation with respect to bases and greedy approximation with respect to redundant systems (dictionaries). Lebesgue-type inequalities for greedy approximation are of special interest.
• Compressed sensing. Exact and approximate recovery by l1 minimization and greedy methods. Incoherent systems. Compressed sensing in Banach spaces.
• Learning theory. Mathematical aspects of supervised learning. Application of approximation theory and greedy approximation in regression and classification problems.
• Numerical integration. Multivariate numerical integration. Discrepancy theory. Fibonacci cubature formulas. Sparse grids.

### Current Projects

• Greedy expansions in Hilbert Spaces. Jointly with J. Nelson we study rate of convergence of expansions of elements in a Hilbert space $H$ into series with regard to a given dictionary $D$. Our primary goal is to study representations of an element $f\in H$ by a series

$f\sim \displaystyle\sum\limits _ {j=1}^\infty c _ j(f)g _ j(f), \quad g _ j(f) \in D.$

In building such a representation we should construct two sequences: $\{g _ j(f)\} _ {j=1}^\infty$ and $\{c _ j(f)\} _ {j=1}^\infty$. We consider construction of $\{g _ j(f)\} _ {j=1}^\infty$ that is based on ideas used in greedy-type nonlinear approximation. This explains the use of the term greedy expansion.
Known algorithms provide rate of convergence of greedy expansions of $f\in A _ 1(D)$ slower than $m^{-1/4}$. It is an interesting open problem: What is the best possible rate of convergence of greedy expansions for $f\in A _ 1(D)$? Our recent qualitative result is that the best possible rate of convergence in the above problem is faster than $m^{-1/4}$. We proved that it is faster than $m^{-2/7}$.

• Lebesgue-type inequalities for greedy approximation in Banach spaces. Jointly with D. Savu we study sparse representations and sparse approximations with respect to incoherent dictionaries. We address the problem of designing and analyzing greedy methods of approximation. A key question in this regard is: How to measure efficiency of a specific algorithm? Answering this question we prove the Lebesgue-type inequalities for algorithms under consideration. A very important new ingredient is that we perform our analysis in a Banach space instead of a Hilbert space. It is known that in many numerical problems users are satisfied with a Hilbert space setting and do not consider a more general setting in a Banach space. There are known arguments that justify interest in Banach spaces. We give one more argument in favor of consideration of greedy approximation in Banach spaces. We introduce a concept of $M$-coherent dictionary in a Banach space which is a generalization of the corresponding concept in a Hilbert space. We analyze the Quasi-Orthogonal Greedy Algorithm (QOGA), which is a generalization of the Orthogonal Greedy Algorithm (Orthogonal Matching Pursuit) for Banach spaces. It is known that the QOGA recovers exactly $S$-sparse signals after $S$ iterations provided $S<(1+1/M)/2$. This result is well known for the Orthogonal Greedy Algorithm in Hilbert spaces. The following question is of great importance: Are there dictionaries in $\mathbb{R}^n$ such that their coherence in $\ell _ p^n$ is less than their coherence in $\ell _ 2^n$ for some $p\in (1,\infty)$? We show that the answer to the above question is "yes". Thus, for such dictionaries, replacing the Hilbert space $\ell _ 2^n$ by a Banach space $\ell _ p^n$ we improve an upper bound for sparsity that guarantees an exact recovery of a signal.

## Teaching Activities

### Previous Courses

• MATH 241 - Vector Calculus
• MATH 550 - Vector Analysis
• MATH 703 - Analysis I
• MATH 704 - Analysis II
• MATH 725 - Approximation Theory
• MATH 728C - Selected Topics: Compressed Sensing
• MATH 728G - Selected Topics: Greedy Approximation
• MATH 728L - Selected Topics: Learning Theory
• MATH 728N - Selected Topics: Numerical Integration
• MATH 729 - Nonlinear Approximation Theory

## Honors and Other Special Scientific Recognition

• USC Educational Foundation Award for Research in Science, Mathematics and Engineering, 2003.
• Invited Hour Speaker, Foundations of Computational Mathematics, Santander, Spain, July 8, 2005.
• Invited Hour Speaker, Curves and Surfaces, Avignon, France, June 29, 2006.
• Invited forty-five minutes lecture in the section Analysis of the International Congress of Mathematicians, Madrid, Spain, August 2006.
• Carolina Distinguished Professor, August 15, 2007.

## 5 Selected Publications

• D. Donoho, M. Elad, and V.N. Temlyakov, On the Lebesgue type inequalities for greedy approximation, J. Approximation Theory 147 (2007), 185-195.
• V.N. Temlyakov, Greedy approximation, Acta Numerica (2008), Cambridge University Press, 235-409.
• V.N. Temlyakov, Mingrui Yang, and Peixin Ye, Greedy approximation with regard to non-greedy bases, Adv. Comput. Math. 34 (2011), 319-337.
• B.S. Kashin and V.N. Temlyakov, A remark on compressed sensing, Matem. Zametki 13 (2007), 71-86.
• V.N. Temlyakov, Greedy Approximation, Cambridge University Press, Cambridge, 2011.