Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Geometric Interpretation of Running Nystr\"{o}m-Based Kernel Machines
  and Error Analysis   Recently, Nystr\"{o}m method has proved its prominence empirically and
theoretically in speeding up the training of kernel machines while retaining
satisfactory performances and accuracy. So far, there are several different
approaches proposed to exploit Nystr\"{o}m method in scaling up kernel
machines. However, there is no comparative study over these approaches, and
they were individually analyzed for specific types of kernel machines.
Therefore, it remains a question that the philosophy of which approach is more
promising when it extends to other kernel machines. In this work, motivated by
the column inclusion property of Gram matrices, we develop a new approach with
a clear geometric interpretation for running Nystr\"{o}m-based kernel machines.
We show that the other two well-studied approaches can be equivalently
transformed to be our proposed one. Consequently, analysis established for the
proposed approach also works for these two. Particularly, our proposed approach
makes it possible to develop approximation errors in a general setting.
Besides, our analysis also manifests the relations among the aforementioned two
approaches and another naive one. First, the analytical forms of the
corresponding approximate solutions are only at odds with one term. Second, the
naive approach can be implemented efficiently by sharing the same training
procedure with others. These analytical results lead to the conjecture that the
naive approach can provide more accurate approximate solutions than the other
two sophisticated approaches. Since our analysis also offers ways for computing
the accuracy of these approximate solutions, we run experiments with
classification tasks to confirm our conjecture.

27
The categories of this paper are: comp-gas
4
The categories of this paper are: cond-mat
23
The categories of this paper are: solv-int
7
The categories of this paper are: hep-th
21
The categories of this paper are: adap-org
34
The categories of this paper are: chem-ph
6
The categories of this paper are: astro-ph
26
The categories of this paper are: bayes-an
33
The categories of this paper are: plasm-ph
13
The categories of this paper are: quant-ph
36
The categories of this paper are: cmp-lg
3
The categories of this paper are: physics
5
The categories of this paper are: gr-qc
28
The categories of this paper are: alg-geom
9
The categories of this paper are: nlin
35
The categories of this paper are: mtrl-th