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XRDS • SUMMER 2014 • VOL. 20 • NO. 4 65
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Exploring Data with Topological Tools

BY MARINKA ZI TNIK
HELLO WORLD

Tools from topology increasingly serve to inspire the development of novel computational methods for data analysis [ 1, 2, 3]. With these methods we can study qualitative geometric information of the data to understand how they are organized on a large scale, and focus on intrinsic shape properties rather than on characteristics that depend on a particular choice of a coordinate system. Here, we apply a topological tool called Mapper [ 1, 2] to extract and visualize simple descriptions of a data set of handwritten digits.

Obtaining Data

We consider the well-known, handwritten digits data set [ 4]. It consists of bitmaps of handwritten digits gathered from a number of people. The 32x32 bitmaps were subdivided into nonoverlapping blocks of 4x4 and the number of pixels were counted in each block. The final data points are represented by 8x8 matrices. In our analysis we utilize the set, which contains 1,797 data points. The preprocessed set is available for download from the UCI Machine Learning Repository (http://archive. ics.uci.edu/ml). In the following code snippets we assume the data is given by a 1797x64 matrix, X, that holds vectorized representation of digits and a vector of integers i range {0, 9}, called targets, corresponding to digits of the data points.

Extracting Data Patterns with Mapper

Mapper is a recently introduced tool for exploratory data analysis that is based on ideas from topology [ 2]. It can be used to reduce a high-dimensional data set into a discrete combinatorial object with far fewer data points, which describe the shape characteristics of complex data at a specified resolution.

For the purpos of this column we
use the MATLAB implementation of
Mapper provided by the authors of the
method, which is available for download
from http://comptop.stanford.edu.

Mapper takes as its input a matrix
of inter-point distances between
data points and a function whose
alue should be defined for all data
points. This function is called a filter
function. It is used to find a covering
of a given data set and for dividing it
into smaller subsets determined by
decomposing filter’s range into a set of
smaller overlapping intervals. Mapper’s
behavior is governed by two parameters,
resolution and percent overlap,
which control the resolution of the
generated data summarization. The two
parameters define the length of each of
the filter’s interval and the percentage
overlap between successive intervals,
respectively. The distance metric and
filter function used in the analysis are
Euclidean and principal SVD values,
The algorithm of Mapper involves
three key steps. First, the filter function
is applied to determine the overlapping
intervals. Then every subset of data
points associated with an interval is
clustered separately. Finally, Mapper
incorporates visualization by providing
an abstract simplicial complex [ 5] that
can be used to further explore the data.
In our analysis of digits data set, the
complex reduces to a network because
we use a single one-dimensional filter
function. Formally, we get as output a
complex in which the highest dimension
of simplices is one. In our case these
are edges in a network. The nodes
correspond to clusters of data points
and a pair of nodes is connected when
corresponding data collections have one
or more data points in common.

Definition 1: A MATLAB script to compute the distance matrix between bitmaps, specify the filter function for decomposing data space and set Mapper parameters.

d = L2_distance(X’, X’, 1);

[U, S, V] = svd(X); filter = U(:, 2)’; filterSamples = 15; overlapPct = 50;

[adja, nodeInfo, levelIdx] = mapper(d, filter, 1/filterSamples, overlapPct);

Definition 2: A MATLAB script to prepare Mapper’s output for network visualization. Mapper’s script writeDotFile writes an adjacency matrix in the DOT format for use with Graphviz.

for i=1:length(nodeInfo)
ecc(i) = nodeInfo{i}.filter;
setSize(i) = length(nodeInfo{i}.set);
end
for i=1:max(size(adja))
digs = nodeInfo{i}.set;
nums = targets(digs);
counts = arrayfun(@(x)sum(nums==x), [0: 9]);
[y, ind] = max(counts);
nodeLabels{i} = ind- 1;
end
writeDotFile(sprintf(‘ digits.dot’), adja, ecc, setSize, {}, nodeLabels);