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Building a sketch of an n-by-n empirical kernel matrix is a common approachto accelerate the computation of many kernel methods. In this paper, we proposea unified framework of constructing sketching methods in kernel ridgeregression (KRR), which views the sketching matrix S as an accumulation of mrescaled sub-sampling matrices with independent columns. Our frameworkincorporates two commonly used sketching methods, sub-sampling sketches (knownas the Nystr\"om method) and sub-Gaussian sketches, as special cases with m=1and m=infinity respectively. Under the new framework, we provide a unifiederror analysis of sketching approximation and show that our accumulation schemeimproves the low accuracy of sub-sampling sketches when certain incoherencecharacteristic is high, and accelerates the more accurate but computationallyheavier sub-Gaussian sketches. By optimally choosing the number m ofaccumulations, we show that a best trade-off between computational efficiencyand statistical accuracy can be achieved. In practice, the sketching method canbe as efficiently implemented as the sub-sampling sketches, as only minor extramatrix additions are needed. Our empirical evaluations also demonstrate thatthe proposed method may attain the accuracy close to sub-Gaussian sketches,while is as efficient as sub-sampling-based sketches.

Accumulations of Projections--A Unified Framework for Random Sketches in Kernel Ridge Regression