Ken Clarkson’s Talk at FWCG 2015

I was extremely fortunate to attend Ken Clarkson‘s excellent talk on “Input Sparsity and Hardness for Robust Subspace Approximation” at FWCG this year. Excerpts from the talk –

Let’s assume that we are given an input of n points a_1, a_2, ... , a_n in d dimensions. These points are arranged as rows of a matrix A \in \mathbb{R}^{n * d}.

The goal is to find a k – dimensional subspace minimizing cost(A,L) \equiv \sum_{i\in [n]} dist(a_i,L)^2 among all possible k-dimensional subspaces.

The immediate solution is through Singular Value Decomposition(SVD hereon) – there is a best rank-k A_k = U_k\sum_kV_k^\top with cost(A,L^*) = \|A - A_k\|_F^2 = \sum_{i\in[n]}dist(a_i, a_iV_kV_k^\top)^2 where L^* is the span of top k right singular vectors in V_k^\top. The best known approximation algorithms can find L' such that dim(L') = k and cost(A, L') \le (1 + \epsilon)cost(A, L^*) in O(nnz(A) + (n + d)poly(k\epsilon^{-1})). It is also important to note that SVD primarily functions by reducing the dimensionality of the given set of data points and hence, a single point (or an outlier) can carry a lot of weightage and influence the decisions made in the decomposition. This leads to the conclusion that SVD is not a very robust technique.

The alternative solution is to use M-Estimators for Regression – Let r_i's be the residuals i.e the difference between the ith observation and the fitted value. The traditional least squares method tries to minimize \sum_i{r_i^2} which runs into trouble when there are outliers in the data as we saw in the earlier case. The M-estimators uses \min_x\sum_iM(r_i) to address this issue and are proven to reduce the effect of outliers.

Now that a more robust technique is established, we need to see how it can be applied to low-rank approximation. Low-rank approximation also calls for rotational invariance where we want the cost to be preserved even if \mathbb{R}^d is rotated by a matrix W. For example, cost functions of the form \sum_i{|a_i - b_i|}_1 can never be used since when B has a zero rank, the rotational invariance property is not preserved for most rotations. The workaround is to use cost functions of the form \sum_{i \in [n]}M(dist(a_i, L)) for M(z) = z^p.  These functions are rotational invariant and more robust for smaller values of p. For any basis V of L, we have cost(A,L) = cost(A,V) \equiv \sum_{i\in [n]} M(dist(a_i, a_i V V^\top)).  A subspace L' is a (1 + \epsilon) approximation if dim(L') = k and cost(A, L') <= (1 + \epsilon)min_{dim(L) = k}cost(A, L).

Some nice results on this front –

  1. Hardness – It is NP-hard find a O(1 + 1/d) approximation for p \in [1,2).
  2. For p \in [1, 2), an algorithm exists such that it returns a (1 + \epsilon)-approximation in O(nnz(A) +(n+d)poly(k\epsilon^{-1}) +exp(poly(k\epsilon^{-1}))) with constant probability.
  3. Results for M-estimators that are nice –
    1. All such functions are even – M(z) = M(-z)
    2. Monotonic – |a| > |b| -> M(a) > M(b)
    3. Bounded growth – There is a C_M such that for a \ge b > 0, C_Ma/b \le M(a)/M(b) \le a^2/b^2.
    4. Root sub-additivity property – \sqrt{M(a + b)} \le \sqrt{M(a)} + \sqrt{M(b)}.
  4. For any vector x, \sqrt{\sum_{i \in [n]}M(x_i)} is almost a norm except for a few properties like scale invariance, scale insensitivity, etc.

There are some open questions too –

  1. If \epsilon is fixed, what is the hardness of approximation?

He also went onto describe some algorithms for the nice M-estimators and row and column reduction techniques which preserve the approximation. Overall, I would say it was a wonderful talk which helped me understand low rank approximations.

Note – I have only sketched a basic idea on the topic here. I strongly recommend the following papers for further reading –

1. Sparse Subspace Embeddings

2. Sketching for M-estimators.

Acknowledgements – I would like to thank Ken Clarkson for his support and encouragement(and slides :)), which played a major role in this blog post.

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