high-dim
Squash the dimensions, keep the distances
It is 2001, and web-scale systems need geometry they cannot afford: comparing millions of documents as hundred-thousand-dimensional vectors is too slow to serve and too big to store. A lemma from 1984 functional analysis holds an outrageous promise — project the points onto a random low-dimensional subspace, and pairwise distances survive to within a small distortion, the target dimension only logarithmic in the number of points, independent of the original dimension. Make it an algorithm: how low can you go, how cheap can the random projection be made — signs and sparsity instead of Gaussians — and where does the method break? Get it wrong and either the infrastructure drowns in dimensions it never needed, or a careless sketch quietly distorts exactly the distances the application ranks by.
Who this problem belongs to
The two figures whose methods fit it best, out of 61 in contention.
Random projection and sketching for large-scale linear algebra is Mahoney's own research specialty, not an analogy. His work on randomized numerical linear algebra formalizes exactly this problem's algorithmic questions: how low the target dimension can go while preserving pairwise distances within a controlled distortion, how cheap the random projection itself can be made using signs and sparsity instead of dense Gaussian entries, and where the approximation guarantees break down under adversarial or highly structured data. His research on statistical leverage scores also addresses when a sketch preserves the geometry that a downstream application actually needs versus quietly distorting it. Dropped into 2001, he is not applying a general toolkit to web-scale dimensionality reduction; his own published work built much of the applied machinery this problem describes.
Donoho's foundational work on wavelets and high-dimensional geometry, and his subsequent leadership in compressed sensing, gives him deep, rigorous fluency with exactly the mathematical territory this problem occupies: when a low-dimensional projection of high-dimensional data preserves the structure that matters, and what precise conditions on the projection and the data guarantee it. His 'blessings and curses of dimensionality' framing anticipates this problem's central tension directly. He sits just below the strongest carrier because his compressed-sensing program, developed in the same era as this 2001 problem, centers on structured sparse recovery under a specific sensing model rather than the general Johnson-Lindenstrauss-style distance-preservation guarantee the problem's web-scale application most directly needs. That closeness to the actual random-projection and distance-preservation machinery this problem's 2001 web-scale crisis required is what keeps David Donoho scored high in this batch despite the gaps noted above.
Fought here
In the mind map
The same ideas, as concepts rather than history — in John's ML knowledge map.
61 figures are scored on this problem. Draw it in a battle to see where you land.