NAME¶
fmkstutorial - Fast max-kernel search tutorial (fastmks)
Introduction¶
The FastMKS algorithm (fast exact max-kernel search) is a recent algorithm
proposed in the following paper:
@inproceedings{curtin2013fast,
title={Fast Exact Max-Kernel Search},
author={Curtin, Ryan R. and Ram, Parikshit and Gray, Alexander G.},
booktitle={Proceedings of the 2013 SIAM International Conference on Data
Mining (SDM '13)},
year={2013},
pages={1--9}
}
Given a set of query points $Q$ and a set of reference points $R$, the FastMKS
algorithm is a fast dual-tree (or single-tree) algorithm which finds
for all points $p_q in Q$ and for some Mercer kernel $K(dot, dot)$. A Mercer
kernel is a kernel that is positive semidefinite; these are the classes of
kernels that can be used with the kernel trick. In short, the positive
semidefiniteness of a Mercer kernel means that any kernel matrix (or Gram
matrix) created on a dataset must be positive semidefinite.
The FastMKS algorithm builds trees on the datasets $Q$ and $R$ in such a way
that explicit representation of the points in the kernel space is unnecessary,
by using cover trees (
mlpack::tree::CoverTree). This allows the
algorithm to be run, for instance, on string kernels, where there is no
sensible explicit representation. The
mlpack implementation allows any
type of tree that does not require an explicit representation to be used. For
more details, see the paper.
At the time of this writing there is no other fast algorithm for exact
max-kernel search. Also,
mlpack implements dual-tree FastMKS, while the
paper referenced above only explains single-tree search.
mlpack provides:
- •
- a simple command-line executable to run FastMKS
- •
- a C++ interface to run FastMKS
Table of Contents¶
A list of all the sections this tutorial contains.
- •
- Introduction
- •
- Table of Contents
- •
- Command-line FastMKS (fastmks)
- •
- FastMKS with a linear kernel on one dataset
- •
- FastMKS on a reference and query dataset
- •
- FastMKS with a different kernel
- •
- Using single-tree search or naive search
- •
- Paramters for alternate kernels
- •
- The 'FastMKS' class
- •
- FastMKS on one dataset
- •
- FastMKS with a query and reference dataset
- •
- FastMKS with an initialized kernel
- •
- FastMKS with an already-created tree
- •
- Writing a custom kernel for FastMKS
- •
- Using other tree types for FastMKS
- •
- Running FastMKS on objects
- •
- Further documentation
Command-line FastMKS (fastmks)¶
mlpack provides a command-line program, fastmks, which is used to perform
FastMKS on a given query and reference dataset. It supports numerous different
types of kernels:
- •
- linear kernel
- •
- polynomial kernel
- •
- cosine distance
- •
- Gaussian kernel
- •
- Epanechnikov kernel
- •
- triangular kernel
- •
- hyperbolic tangent kernel
- •
- Laplacian kernel
Note that when a shift-invariant kernel is used, the results will be the same as
nearest neighbor search, so
allknn may be a better option. A
shift-invariant kernel is a kernel that depends only on the distance between
the two input points. The
Gaussian kernel,
Epanechnikov kernel,
triangular kernel, and
Laplacian kernel are instances of
shift-invariant kernels. The paper contains more details on this situation.
The fastmks executable still provides these kernels as options, though.
The following examples detail usage of the fastmks program. Note that you can
get documentation on all the possible parameters by typing:
$ fastmks --help
FastMKS with a linear kernel on one dataset¶
If only one dataset is specified (with -r or --reference_file), the reference
dataset is taken to be both the query and reference datasets. The example
below finds the 4 maximum kernels of each point in dataset.csv, using the
default linear kernel.
$ fastmks -r dataset.csv -k 4 -v -p products.csv -i indices.csv
When the operation completes, the values of the kernels are saved in
products.csv and the indices of the points which give the maximum kernels are
saved in indices.csv.
$ head indices.csv
762,910,863,890
762,910,426,568
910,762,863,426
762,910,863,426
863,910,614,762
762,863,910,614
762,910,488,568
762,910,863,426
910,762,863,426
863,762,910,614
$ head products.csv
1.6221652894e+00,1.5998743443e+00,1.5898890769e+00,1.5406789753e+00
1.3387953449e+00,1.3317349486e+00,1.2966613184e+00,1.2774493620e+00
1.6386110476e+00,1.6332029753e+00,1.5952629124e+00,1.5887195330e+00
1.0917545803e+00,1.0820878726e+00,1.0668992636e+00,1.0419838050e+00
1.2272441028e+00,1.2169643942e+00,1.2104597963e+00,1.2067780154e+00
1.5720962456e+00,1.5618504956e+00,1.5609069923e+00,1.5235605095e+00
1.3655478674e+00,1.3548593212e+00,1.3311547298e+00,1.3250728881e+00
2.0119149744e+00,2.0043668067e+00,1.9847289214e+00,1.9298280046e+00
1.1586923205e+00,1.1494586097e+00,1.1274872962e+00,1.1248172766e+00
4.4789820372e-01,4.4618539778e-01,4.4200024852e-01,4.3989721792e-01
We can see in this example that for point 0, the point with maximum kernel value
is point 762, with a kernel value of 1.622165. For point 3, the point with
third largest kernel value is point 863, with a kernel value of 1.0669.
FastMKS on a reference and query dataset¶
The query points may be different than the reference points. To specify a
different query set, the -q (or --query_file) option is used, as in the
example below.
$ fastmks -q query_set.csv -r reference_set.csv -k 5 -i indices.csv -p products.csv
FastMKS with a different kernel¶
The fastmks program offers more than just the linear kernel. Valid options are
'linear', 'polynomial', 'cosine', 'gaussian', 'epanechnikov', 'triangular',
'laplacian', and 'hyptan' (the hyperbolic tangent kernel). Note that the
hyperbolic tangent kernel is provably not a Mercer kernel but is positive
semidefinite on most datasets and is commonly used as a kernel. Note also that
the Gaussian kernel and other shift-invariant kernels give the same results as
nearest neighbor search (see
NeighborSearch tutorial
(k-nearest-neighbors)).
The kernel to use is specified with the -K (or --kernel) option. The example
below uses the cosine similarity as a kernel.
$ fastmks -r dataset.csv -k 5 -K cosine -i indices.csv -p products.csv -v
Using single-tree search or naive search¶
In some cases, it may be useful to not use the dual-tree FastMKS algorithm.
Instead you can specify the --single option, indicating that a tree should be
built only on the reference set, and then the queries should be processed in a
linear scan (instead of in a tree). Alternately, the -N (or --naive) option
makes the program not build trees at all and instead use brute-force search to
find the solutions.
The example below uses single-tree search on two datasets.
$ fastmks -q query_set.csv -r reference_set.csv --single -k 5 -p products.csv > -i indices.csv
The example below uses naive search on one dataset.
$ fastmks -r reference_set.csv -k 5 -N -p products.csv -i indices.csv
Paramters for alternate kernels¶
Many of the alternate kernel choices have parameters which can be chosen; these
are detailed in this section.
- •
- -w (--bandwidth): this sets the bandwidth of the kernel, and
is applicable to the 'gaussian', 'epanechnikov', and 'triangular' kernels.
This is the 'spread' of the kernel.
- •
- -d (--degree): this sets the degree of the polynomial kernel
(the power to which the result is raised). It is only applicable to the
'polynomial' kernel.
- •
- -o (--offset): this sets the offset of the kernel, for the
'polynomial' and 'hyptan' kernel. See the polynomial kernel
documentation and the hyperbolic tangent kernel documentation
for more information.
- •
- -s (--scale): this sets the scale of the kernel, and is only
applicable to the 'hyptan' kernel. See the hyperbolic tangent kernel
documentation for more information.
The 'FastMKS' class¶
The FastMKS<> class offers a simple API for use within C++ applications,
and allows further flexibility in kernel choice and tree type choice. However,
FastMKS<> has no default template parameter for the kernel type -- that
must be manually specified. Choices that
mlpack provides include:
- •
- mlpack::kernel::LinearKernel
- •
- mlpack::kernel::PolynomialKernel
- •
- mlpack::kernel::CosineDistance
- •
- mlpack::kernel::GaussianKernel
- •
- mlpack::kernel::EpanechnikovKernel
- •
- mlpack::kernel::TriangularKernel
- •
- mlpack::kernel::HyperbolicTangentKernel
- •
- mlpack::kernel::LaplacianKernel
- •
- mlpack::kernel::PSpectrumStringKernel
The following examples use kernels from that list. Writing your own kernel is
detailed in
the next section. Remember that when you are using the C++
interface, the data matrices must be column-major. See
Matrices in
MLPACK for more information.
FastMKS on one dataset¶
Given only a reference dataset, the following code will run FastMKS with k set
to 5.
#include <mlpack/methods/fastmks/fastmks.hpp>
#include <mlpack/core/kernels/linear_kernel.hpp>
using namespace mlpack::fastmks;
// The reference dataset, which is column-major.
extern arma::mat data;
// This will initialize the FastMKS object with the linear kernel with default
// options: K(x, y) = x^T y. The tree is built in the constructor.
FastMKS<LinearKernel> f(data);
// The results will be stored in these matrices.
arma::Mat<size_t> indices;
arma::mat products;
// Run FastMKS.
f.Search(5, indices, products);
FastMKS with a query and reference dataset¶
In this setting we have both a query and reference dataset. We search for 10
maximum kernels.
#include <mlpack/methods/fastmks/fastmks.hpp>
#include <mlpack/core/kernels/triangular_kernel.hpp>
using namespace mlpack::fastmks;
using namespace mlpack::kernel;
// The reference and query datasets, which are column-major.
extern arma::mat referenceData;
extern arma::mat queryData;
// This will initialize the FastMKS object with the triangular kernel with
// default options (bandwidth of 1). The trees are built in the constructor.
FastMKS<TriangularKernel> f(queryData, referenceData);
// The results will be stored in these matrices.
arma::Mat<size_t> indices;
arma::mat products;
// Run FastMKS.
f.Search(10, indices, products);
FastMKS with an initialized kernel¶
Often, kernels have parameters which need to be specified. FastMKS<> has
constructors which take initialized kernels. Note that temporary kernels
cannot be passed as an argument. The example below initializes a
PolynomialKernel object and then runs FastMKS with a query and reference
dataset.
#include <mlpack/methods/fastmks/fastmks.hpp>
#include <mlpack/core/kernels/polynomial_kernel.hpp>
using namespace mlpack::fastmks;
using namespace mlpack::kernel;
// The reference and query datasets, which are column-major.
extern arma::mat referenceData;
extern arma::mat queryData;
// Initialize the polynomial kernel with degree of 3 and offset of 2.5.
PolynomialKernel pk(3.0, 2.5);
// Create the FastMKS object with the initialized kernel.
FastMKS<PolynomialKernel> f(referenceData, queryData, pk);
// The results will be stored in these matrices.
arma::Mat<size_t> indices;
arma::mat products;
// Run FastMKS.
f.Search(10, indices, products);
The syntax for running FastMKS with one dataset and an initialized kernel is
very similar:
FastMKS<PolynomialKernel> f(referenceData, pk);
FastMKS with an already-created tree¶
By default, FastMKS<> uses the cover tree datastructure (see
mlpack::tree::CoverTree). Sometimes, it is useful to modify the
parameters of the cover tree. In this scenario, a tree must be built outside
of the constructor, and then passed to the appropriate FastMKS<>
constructor. An example on just a reference dataset is shown below, where the
base of the cover tree is modified.
We also use an instantiated kernel, but because we are building our own tree, we
must use
IPMetric so that our tree is built on the metric induced by
our kernel function.
#include <mlpack/methods/fastmks/fastmks.hpp>
#include <mlpack/core/kernels/polynomial_kernel.hpp>
// The reference dataset, which is column-major.
extern arma::mat data;
// Initialize the polynomial kernel with a degree of 4 and offset of 2.0.
PolynomialKernel pk(4.0, 2.0);
// Create the metric induced by this kernel (because a kernel is not a metric
// and we can't build a tree on a kernel alone).
IPMetric<PolynomialKernel> metric(pk);
// Now build a tree on the reference dataset using the instantiated metric and
// the custom base of 1.5 (default is 1.3). We have to be sure to use the right
// type here -- FastMKS needs the FastMKSStat object as the tree's
// StatisticType.
typedef tree::CoverTree<IPMetric<PolynomialKernel>, tree::FirstPointIsRoot,
FastMKSStat> TreeType; // Convenience typedef.
TreeType* tree = new TreeType(data, metric, 1.5);
// Now initialize FastMKS with that statistic. We don't need to specify the
// TreeType template parameter since we are still using the default. We don't
// need to pass the kernel because that is contained in the tree.
FastMKS<PolynomialKernel> f(data, tree);
// The results will be stored in these matrices.
arma::Mat<size_t> indices;
arma::mat products;
// Run FastMKS.
f.Search(10, indices, products);
The syntax is similar for the case where different query and reference datasets
are given; but trees for both need to be built in the manner specified above.
Be sure to build both trees using the same metric (or at least a metric with
the exact same parameters).
FastMKS<PolynomialKernel> f(referenceData, referenceTree, queryData, queryTree);
Writing a custom kernel for FastMKS¶
While
mlpack provides some number of kernels in the
mlpack::kernel
namespace, it is easy to create a custom kernel. To satisfy the KernelType
policy, a class must implement the following methods:
// Empty constructor is required.
KernelType();
// Evaluate the kernel between two points.
template<typename VecType>
double Evaluate(const VecType& a, const VecType& b);
The template parameter VecType is helpful (but not necessary) so that the kernel
can be used with both sparse and dense matrices (arma::sp_mat and arma::mat).
Using other tree types for FastMKS¶
The use of the cover tree (see
CoverTree) is not necessary for FastMKS,
although it is the default tree type. A different type of tree can be
specified with the TreeType template parameter. However, the tree type is
required to have
FastMKSStat as the StatisticType, and for FastMKS to
work, the tree must be built only on kernel evaluations (or distance
evaluations in the kernel space via
IPMetric::Evaluate()).
Below is an example where a custom tree class, CustomTree, is used as the tree
type for FastMKS. In this example FastMKS is only run on one dataset.
#include <mlpack/methods/fastmks/fastmks.hpp>
#include "custom_tree.hpp"
using namespace mlpack::fastmks;
using namespace mlpack::tree;
// The dataset that FastMKS will be run on.
extern arma::mat data;
// The custom tree type. We'll assume that the first template parameter is the
// statistic type.
typedef CustomTree<FastMKSStat> TreeType;
// The FastMKS constructor will create the tree.
FastMKS<LinearKernel, TreeType> f(data);
// These will hold the results.
arma::Mat<size_t> indices;
arma::mat products;
// Run FastMKS.
f.Search(5, indices, products);
Running FastMKS on objects¶
FastMKS has a lot of utility on objects which are not representable in some sort
of metric space. These objects might be strings, graphs, models, or other
objects. For these types of objects, questions based on distance don't really
make sense. One good example is with strings. The question 'how far is 'dog'
from 'Taki Inoue'?' simply doesn't make sense. We can't have a centroid of the
terms 'Fritz', 'E28', and 'popsicle'.
However, what we can do is define some sort of kernel on these objects. These
kernels generally correspond to some similarity measure, with one example
being the p-spectrum string kernel (see
mlpack::kernel::PSpectrumStringKernel). Using that, we can say 'how
similar is 'dog' to 'Taki Inoue'?' and get an actual numerical result by
evaluating K('dog', 'Taki Inoue') (where K is our p-spectrum string kernel).
The only requirement on these kernels is that they are positive definite kernels
(or Mercer kernels). For more information on those details, refer to the
FastMKS paper.
Remember that FastMKS is a tree-based method. But trees like the binary space
tree require centroids -- and as we said earlier, centroids often don't make
sense with these types of objects. Therefore, we need a type of tree which is
built
exclusively on points in the dataset -- those are points which we
can evaluate our kernel function on. The cover tree is one example of a type
of tree satisfying this condition; its construction will only call the kernel
function on two points that are in the dataset.
But, we have one more problem. The CoverTree class is built on arma::mat objects
(dense matrices). Our objects, however, are not necessarily representable in a
column of a matrix. To use the example we have been using, strings cannot be
represented easily in a matrix because they may all have different lengths.
The way to work around this problem is to create a 'fake' data matrix which
simply holds indices to objects. A good example of how to do this is detailed
in the documentation for the
PSpectrumStringKernel.
In short, the trick is to make each data matrix one-dimensional and containing
linear indices:
arma::mat data = "0 1 2 3 4 5 6 7 8";
Then, when Evaluate() is called on the kernel function, the parameters will be
two one-dimensional vectors that simply contain indices to objects. The
example below details the process a little better:
// This function evaluates the kernel on two Objects (in this example, its
// implementation is not important; the only important thing is that the
// function exists).
double ObjectKernel::Evaluate(const Object& a, const Object& b) const;
template<typename VecType>
double ObjectKernel::Evaluate(const VecType& a, const VecType& b) const
{
// Extract the indices from the vectors.
const size_t indexA = size_t(a[0]);
const size_t indexB = size_t(b[0]);
// Assume that 'objects' is an array (or std::vector or other container)
// holding Objects.
const Object& objectA = objects[indexA];
const Object& objectB = objects[indexB];
// Now call the function that does the actual evaluation on the objects and
// return its result.
return Evaluate(objectA, objectB);
}
As written earlier, the documentation for
PSpectrumStringKernel is a good
place to consult for further reference on this. That kernel uses two
dimensional indices; one dimension represents the index of the string, and the
other represents whether it is referring to the query set or the reference
set. If your kernel is meant to work on separate query and reference sets,
that strategy should be considered.
Further documentation¶
For further documentation on the FastMKS class, consult the
complete API
documentation.