# From Wavelets to Compressed Sensing and Beyond

#### 2013 Shaw Prize Recognizes a Remarkable Mathematical Life

On September 23, in formal ceremonies in Hong Kong, David Donoho of Stanford University received the 2013 Shaw Prize in Mathematical Sciences. Citing “his profound contributions to modern mathematical statistics,” the selection committee mentioned in particular his “development of optimal algorithms for statistical estimation in the presence of noise and of efficient techniques for sparse representation and recovery in large data-sets."

He and Johnstone proposed thresholding algorithms that were both efficient and amenable to “detailed and elegant theoretical study using a blend of statistical decision theory, harmonic analysis, and approximation theory.” In estimation theory, Johnstone says, “Donoho changed the paradigm from ‘smoothness’ to ‘sparsity’ by showing that sparsity was the more powerful concept. When sparsity reduced to smoothness, traditional linear methods of estimation such as splines, kernels and spectral cut-off were provably optimal. By contrast, when sparsity could be identified in non-smooth settings, suitable nonlinear methods were demonstrably superior.”

Many in the SIAM community heard about Donoho’s subsequent interests in his invited talk at ICIAM 2003 and in his 2001 SIAM John von Neumann Lecture: “What Lies Beyond Wavelets? Explorations in Multiscale Thinking . . . .” Variants of multiscale thinking woven into the lecture ranged from Fourier integral operators to “ridgelets and curvelets for representation of edges in images, to beamlet detectors for filaments in noisy images.” Applications abounded, in data compression, statistical estimation, and pattern analysis.

Recommended reading for those interested in these research areas, or in the trajectory of such a successful research career, is the “Autobiography” Donoho wrote for the Shaw Prize website. In it, he paid tribute to a very early influence on his career: John Tukey, his undergraduate thesis adviser at Princeton, who advocated “robust statistical methods, such as fitting equations by minimizing the \(\ell_1\) norm of residuals rather than the \(\ell_2\) norm.” Tukey, Donoho wrote, “criticized ‘classical’ mathematical statistics as searching for polished answers to yesterday’s problems.”

In a concluding twist to his autobiographical statement, Donoho commented on a new approach to the sparsity/undersampling tradeoff that he and colleagues had developed: “Solving random underdetermined systems by \(\ell_1\)-minimization was revealed as identical to denoising of sparse signals embedded in noise [his emphasis]. Two separate threads of my research life became unified.”

In the statement, Donoho credited many people, both mathematical forebears and colleagues and students. Sure to play a role in any future autobiographical statement is Run Run Shaw, Hong Kong philanthropist, entertainment tycoon, and co-founder of Shaw Brothers, one of the world’s largest film studios. In 2002 he established the Shaw Prizes, which honor outstanding work in research and applications in the mathematical sciences, life science and medicine, and astronomy; presented annually, the prize in each field carries a cash award of U.S. $1 million. Shaw died in January 2014, at the age of 106.