2016 ICSEE International Conference on the Science of Electrical Engineering

Randomized Independent Component Analysis
Matan Sela

Ron Kimmel

arXiv:1609.06942v1 [stat.ML] 22 Sep 2016

Department of Computer Science
Technion - Israel Institute of Technology

Abstract—Independent component analysis (ICA) is a method
for recovering statistically independent signals from observations
of unknown linear combinations of the sources. Some of the most
accurate ICA decomposition methods require searching for the
inverse transformation which minimizes different approximations
of the Mutual Information, a measure of statistical independence
of random vectors. Two such approximations are the Kernel
Generalized Variance or the Kernel Canonical Correlation which
has been shown to reach the highest performance of ICA
methods. However, the computational effort necessary just for
computing these measures is cubic in the sample size. Hence,
optimizing them becomes even more computationally demanding,
in terms of both space and time. Here, we propose a couple of
alternative novel measures based on randomized features of the
samples - the Randomized Generalized Variance and the Randomized Canonical Correlation. The computational complexity
of calculating the proposed alternatives is linear in the sample
size and provide a controllable approximation of their Kernelbased non-random versions. We also show that optimization of
the proposed statistical properties yields a comparable separation
error at an order of magnitude faster compared to Kernel-based
measures.

I. I NTRODUCTION
Independent component analysis (ICA) is a well-established
problem in unsupervised learning and signal processing, with
numerous applications including blind source separation, face
recognition, and stock price prediction. onsider the following
scenario. A couple of speakers are located in a room. Each
of them plays a different sound . Two microphones which are
arbitrarily placed in the same room record unknown linear
combinations of the sounds. The goal of ICA is to process the
signals recorded by the mics for recovering the soundtracks
played by the speakers.
More precisely, the basic idea of ICA is to recover ns statistically independent components of a non-Gaussian random
T
vector s = (s1 , ..., sns ) from ns observed linear mixtures of
its elements. That is, we assume that some unknown matrix
A ∈ Rns ×ns mixes the entries of s such that x = As. From
samples of x, the goal is to estimate an un-mixing matrix W
such that y = W x and the components of y are statistically
independent.
The matrix W is found by a minimization process over
a contrast function which measures the dependency between
the unmixed elements. Ideally, finding the matrix W which
minimizes the mutual information (MI) provides the theoretically most accurate reconstruction. The MI is defined as the
Kullback-Liebler divergence between the joint distribution of
y, p(y1 , ..., yns ), and the product of its marginal distributions,

ns
Y

p(yi ). It is a non-negative function which vanishes if and

i=1

only if the components of y are mutually independent. Unfortunately, in practical applications, the joint and the marginal
distributions are unknown. Estimating and optimizing the MI
directly from the samples is difficult. Fitting a parametric or a
nonparametric probabilistic model to the data based on which
MI is calculated is problem dependent and is often inaccurate.
Many researches proposed alternative contrast functions.
One of the most robust alternative is the F-correlation proposed in [1]. This function is evaluated by mapping the data
samples into a reproducing kernel Hilbert space, F where
a canonical correlation analysis is performed. The largest
kernel canonical correlation (KCC) and the product of the
kernel canonical correlations, known as the kernel generalized
variance (KGV), are two possible contrast functions. Despite
their superior performance, algorithms based on minimizing
the KCC or the KGV (denoted as kernelized ICA algorithms)
are less attractive for practical uses since the complexity of
exact evaluation of these functions is cubic in the sample size.
A recent strand of research suggested randomized nonlinear
feature maps for approximating the reproducing kernel Hilbert
space corresponding to kernel functions. This technique enables revealing nonlinear relations in a data by performing linear data analysis algorithms, such as Support Vector Machine
[2], Principal Component Analysis and Canonical Correlation
Analysis [3]. These methods approximate the solution of
kernel methods while reducing their complexity from cubic
to linear in the sample size.
Here, we propose two alternative contrast functions, Randomized Canonical Correlation (RCC) and Randomized Generalized Variance (RGV) which approximate KCC and KGV,
respectively, yet require just a fraction of the computational
effort to evaluate. Furthermore, the proposed random approximations are smooth, easy to optimize and converge to their
kernelized version as the number of random features grows.
Finally, we propose optimization algorithms similar to those
proposed in [1], for solving the ICA problem. We demonstrate
that our method has a comparable accuracy as KICA but runs
12 times faster while separating components of real data.
II. BACKGROUND AND R ELATED W ORKS
A. Canonical Correlation Analysis
Canonical correlation analysis is a classical linear analysis
method introduced in [4], which generalizes the Principal
Component Analysis (PCA) for two or more random vectors.

In PCA, given samples of a random vector x ∈ Rd , the
idea is to search for a vector w ∈ Rd , which maximizes
the variance of the projection of x onto w. In practice, the
principal components are the eigenvectors corresponding to
the largest eigenvalues of the empirical covariance matrix
1
XX T , where X is a matrix containing samples of
C =
N
x as its columns, and N is the sample size.
In canonical correlation analysis (CCA), given a couple of
random vectors x ∈ Rdx and y ∈ Rdy , one looks for a
pair of vectors wx ∈ Rdx and wy ∈ Rdy , that maximize
the correlation between the projection of x onto wx and
the projection of y onto wy . More formally, CCA can be
formulated as
max

wx ∈Rdx ,wy ∈Rdy

=

corr(wxT x, wyT y) =

cov(wxT x, wyT y)
(var(wxT x))1/2 (var(wyT y))1/2

Let X ∈ Rdx ×N and Y ∈ Rdy ×N be matrices of samples
of x and y, respectively. The empirical Canonical Correlation
Analysis problem is given by
minimize

Wx ∈Rdx ×r ,Wy ∈Rdy ×r

subject to

kd
corr(WxT X, WyT Y ) − Ik
cd
orr(Wx , Wx ) = I,
cd
orr(Wy , Wy ) = I,

where cd
orr(·, ·) is the empirical correlation. The canonical
correlations λ1 , ..., λr are found by solving the following
generalized eigenvalue problem



0
CXY
wx
=
CY X
0
wy



CXX + γI
0
wx
λ
,
0
CY Y + γI
wy
1
XY T is the empirical cross covariance
where CXY =
N
matrix, and γI is added to the diagonal for stabilizing the
solution. As discussed next, the kernelized ICA method use a
kernel formulation of CCA for evaluating its contrast functions
which measures the independence of random variables.
B. Kernelized Independent Component Analysis
In ICA, we minimize a contrast function which is defined
as any non-negative function of two or more random variables
that is zero if and only if they are statistically independent.
By definition, a pair of random variables x1 and x2 are said
to be statistically independent if p(x1 , x2 ) = p(x1 )p(x2 ). As
a result, for any two functions f1 , f2 ∈ F

The F-correlation ρF is defined as the maximal correlation
among all the functions f1 (x1 ), f2 (x2 ) in F. That is

=

f1 ,f2 ∈F

|cov (f1 (x1 ), f2 (x2 ))|

max

f1 ,f2 ∈F

1/2

(varf1 (x1 ))

(varf2 (x2 ))

1/2

Obviously, if x1 and x2 are independent, then ρF = 0. As
proven in [1], if F is a functional space that contains the
Fourier basis (f (x) = eiωx , ω ∈ R), then, the opposite is also
true. This property implies that ρF can replace the mutual
information while searching for a matrix W that transforms the
vector (x1 , x2 )T into a vector with independent components.
Computing the F-correlation directly for an arbitrary space
F requires estimating the correlation between every possible
pair of functions in the space, making the calculation impractical. However, if F is a reproducing kernel Hilbert space
(RKHS), the F-correlation can be evaluated. Let k(x, y) be
a kernel function associated with the inner product between
functions in a reproducing kernel Hilbert space F. Denote
Φ(x) as the feature map corresponding to the kernel k(x, y),
such that k(x, y) = hΦ(x), Φ(y)i. The feature map is given
by Φ(x) = k(·, x). Then, from the reproducing property of
the kernel, for any function f (x) ∈ F,
f (x) = hΦ(x), f i,

∀f ∈ F, ∀x ∈ R.

It follows that for every functional space F associated with
a kernel function k(x, y) and a feature map Φ(x), the Fcorrelation between a pair of random variables x1 and x2 can
be formulated as
ρF = max |corr (hΦ(x1 ), f1 i, hΦ(x2 ), f2 i)| .
f1 ,f2 ∈F

i N
Let {xi1 }N
i=1 and {x2 }i=1 be the samples of the random
variables x1 and x2 , respectively. Any function f ∈ F
N
X
can be represented by f =
αi Φ(xi ) + f ⊥ , where f ⊥
i=1

is a function in the subspace of functions orthogonal to
span{Φ(x1 ), ..., Φ(xN )}. Thus,
c (hΦ(x1 ), f1 i, hΦ(x2 ), f2 i) =
cov
N
1 X
hΦ(xk1 ), f1 ihΦ(xk2 ), f2 i
=
N
k=1

N
N
N
X
X
1 X
=
hΦ(xk1 ),
α1i Φ(xi1 ) + f1⊥ ihΦ(xk2 ),
α2i Φ(xi2 ) + f2⊥ i
N
i=1
i=1

=

E{f1 (x1 )f2 (x2 )} = E{f1 (x1 )}E{f2 (x2 )}

1
N

k=1
N
X

hΦ(xk1 ),

N
X

α1i Φ(xi1 )ihΦ(xk2 ),

N
X

α2i Φ(xi2 )i

i=1
i=1
k=1
N X
N X
N
X
α1i K1 (xi1 , xk1 )K2 (xj2 , xk2 )α2i
k=1 i=1 j=1

=

1
N

=

1 T
α K1 K2 α2 ,
N 1

Equivalently,
cov (f1 (x1 ), f2 (x2 )) = 0.

max |corr (f1 (x1 ), f2 (x2 ))|

ρF =

where K1 and K2 are the empirical kernel Gram matrices of
x1 and x2 , respectively. With a similar development for the
empirical variance, one conclude
c (hΦ(x1 ), f1 i) =
var
1 T
α K1 K1 α1
N 1
c (hΦ(x2 ), f2 i) =
var
1 T
α K2 K2 α2
N 2
Therefore, the empirical F-correlation between x1 and x2
is given by
ρ̂F (x1 , x2 ) =

max

α1 ,α2 ∈RN

where P is the rank of Kκ . Figure II-B demonstrates the
Kernel Generalized Variance versus the Mutual Information
for a Gaussian kernel and for different σ’s. The complexity of
a naive implementation of both KCC and KGV is O(N 3 ).
However, an approximate solution can be computed using
Cholesky decomposition which reduces the complexity to be
roughly quadratic in the sample size.

α1T K1 K2 α2
1/2 T 2 1/2
α2 K2 α2
α1T K12 α1

which can be evaluated by solving the following generalized
eigenvalue problem



0
K1 K2
α1
=
K2 K1
0
α2
 2


K1
0
α1
λ
.
0 K22
α2
The calculation of the F-correlation is thus equivalent to
solving a kernel CCA problem. For ensuring computational
stability, it is common to use a regularized version of KCCA
given by



0
K1 K2
α1
=
K2 K1
0
α2




0
α1
(K1 + N2κ I)2
λ
.
α2
0
(K2 + N2κ I)2
For m random variables, the regularized KCCA amounts
to finding the largest generalized eigenvalue of the following
system



0

 K2 K1

..

.
Km K1



λ


K1 K2
0
..
.
Km K2

(K1 + N2κ I)2
0
..
.
0

···
···

···

α1
K1 Km
K2 Km   α2 
 .  =
..
  .. 
.
0
αm

0
(K2 + N2κ I)2
..
.
0

···
···
···

P

for different σ values as a function of the angle of rotation of the
orthogonal matrix W .

C. Randomized Features
In kernel methods, each data sample x is mapped to a
function in the reproducing kernel Hilbert space Φ(x), where
the analysis is performed. The inner product between two
representational functions Φ(x) and Φ(y) can be evaluated
directly on the samples x and y using the kernel function
k(x, y). For real valued, normalized (k(x, y) ≤ 1), shift
invariant kernels Rd × Rd
Z
T
k(x, y) =
p(w)e−jw (x−y) dw ≈
Rd





0
α1
0
  α2 
 . 
..
  .. 
.
2
Nκ
αm
(Km + 2 I)

or in short, Kκ α = λDκ α.
This is the first contrast function proposed in [1]. The
second one, denoted as the kernel generalized variance, depends upon not only the largest generalized eigenvalue of the
problem above but also on the product of the entire spectrum.
This is derived from the Gaussian case where the mutual
information is equal to minus half the logarithm of the product
of generalized eigenvalues of the regular CCA problem. In
summary, the kernel generalized variance is given by
I(x1 , ..., xm ) = −

Fig. 1. The mutual information and the Kernel Generalized Variance

1X
log λi ,
2 i=1

m
X
1 −jwiT x jwiT y
≈
e
e
m
i=1

=

m
X
1
cos(wiT x + bi ) cos(wiT y + bi ),
m
i=1

where p(w) is the inverse Fourier transform of k and bi ∼
U (0, 2π), and wi ’s are independently drawn from p(w). Thus,
the kernel function can be approximated by transforming the
data samples into an m-dimensional random space z(x) =

1 
T
√
cos(w1T x + b1 ), ..., cos(wm
x + bm ) and taking the inm
ner product between the maps.
k(x, y) ≈ hz(x), z(y)i.
The following theorem shows that the reproducing kernel
Hilbert space corresponding to k can be approximated by a
random map, in the inner product sense.

Let k(x, y) = k(x − y) be a shift-invariant kernel function
such that k(x, y) ≤ 1. Denote p(w)
as the inverse Fourier
Z
transform of k, that is, k(x) =

p(w)e−i2πw

T

x

dx. Inde-

Rd
{wi }m
i=1

pendently sample m variables
from the distribution
p(w) and m random numbers bi uniformly from the section
[−π,
r π] and construct the r andom Fourier feature map z(x) =

2 
T
cos(w1T x + b1 ), ..., cos(wm
x + bm ) . We define the Zm
correlation of a pair of random variables x1 and x2 as
ρz (x1 , x2 ) =

max

w1 ∈Rm ,w2 ∈Rm

corr(w1T z(x1 ), w2T z(x2 ))

The empirical covariance between w1T z(x1 ) and w2T z(x2 )
is given by
c 1T z(x1 ), w2T z(x2 )) =
cov(w

Fig. 2. The analytic and empirical error of the kernel approximation

using random Fourier features as a function of the number of features
m.

N
1 X
hw1 , z(xk1 )ihw2 , z(xk2 )i
=
N
k=1

N
1 X T
w1 z(xk1 )z(xk2 )T w2
=
N
k=1
!
N
X
1
= w1T
z(xk1 )z(xk2 )T w2
N

d×N

Theorem II.1. Let X ∈ R
be a matrix containing samples
of x ordered in its columns. Denote z(X) ∈ Rm×N as a
matrix containing the Fourier random features of each column
of X as its columns. Denote K̂ = z(X)T z(x) and K as the
empirical kernel matrix corresponding to the same kernel as
K̂. Then
r
3n2 log n 2n log n
EkK̂ − Kk ≤
+
,
m
m

k=1

z
= w1T Ĉ12
w2

Similarly, the empircal variances can be computed as
z
vd
ar(w1T z(x1 )) = w1T Ĉ11
w1

where the norm is an operator norm.

z
vd
ar(w2T z(x2 )) = w1T Ĉ22
w2 ,

Proof. See [3].
Figure II-C shows the analytic bound versus the empirical
error of an approximate kernel K̂ evaluated on 104 data
samples.
Introduced in [2] for approximating the solution of kernel
Support Vector Machine, these random feature maps are
also useful for solving kernel Principal Component Analysis
(kPCA), and kernel Canonical Correlation Analysis (kCCA),
[3]. The main purpose of using these features is the fact that
they enable reducing the complexity of kernel methods to
linear in the sample size, at the expense of a mild gap in
accuracy. Here, we extend this idea for solving the problem
of ICA.
III. R ANDOMIZED I NDEPENDENT C OMPONENT A NALYSIS
As demonstrated in [1], the kernel canonical correlation
and the kernel generalized variance measure the statistical
dependence between a set of sampled random variables. In
addition, optimization over these functions for decomposing
mixtures of these variables into independent ones is more accurate and robust. However, evaluating these functions requires
O(N 3 ) operations. Fortunately, for certain types of kernels,
these contrast functions can be approximated using random
features in linear time.

where

z
Ĉ11

=

N
1 X
z(xk1 )z(xk1 )T
N

and

z
Ĉ22

=

k=1

N
1 X
z(xk2 )z(xk2 )T .
N
k=1
Thus the empirical Z-correlation is the largest generalized
eigenvalue of the system



z
0
Ĉ12
w1
=
z
w
Ĉ21
0
 z
 2 
Ĉ11 + γI
0
w1
λ
.
(1)
z
w2
+ γI
0
Ĉ22

As demonstrated in Figure 3, as the number of random features grows, the Z-correlation converges to the F-correlation.
Notice that the size of the generalized eigenvalue system in the
random feature space is ns m × ns m which is much smaller
than the kernel case where it is of size ns N × ns N .
The kernel canonical correlation provide less accurate results than the kernel generalized variance since it takes into
account only the largest generalized eigenvalue. This can also
be justified by the Gaussian case where the kernel generalized
variance is shown to be a second order approximation of the
mutual information around the point of independence as σ

and comparing the accuracy of our algorithm in estimating
the unmixing matrix from the mixed data, we used the Amari
distance given by
!
Pns
ns
1 X
j=1 |aij |
d(V, W ) =
−1 +
2ns i=1 maxj |aij

ns  Pns
1 X
i=1 |aij |
+
−1 ,
2ns j=1 maxi |aij |
where aij = (V W −1 )ij . This metric, introduced in [5], is
invariant to scaling and permutation of the rows or columns
of the matrices.

Fig. 3. Approximation of the Kernel Canonical Correlation using Ran-

domized Canonical Correlation with different numbers of features.

goes to zero. Thus, we propose to approximate the kernel
generalized variance by
P

δz (x1 , ..., xns ) = −

1X
log λk
2
k=1

where λk are the generalized eigenvalues of 1. We define
this function as the random generalized variance (RGV). As
demonstrated in Figure 4, the more features we take for
calculating the empirical covariances in the random space, the
better RGV approximates KGV.
Fig. 5. Probability density functions used in our tests. These pdf-s

are identical to those used in [1].

Approximation of the Kernel Generalized Variance using
Randomized Generalized Variance with different numbers of features.
Fig. 4.

IV. E XPERIMENTAL R ESULTS
To evaluate the performance of optimization over our novel
contrast functions for solving ICA problems, we independently
draw samples from two or more probability functions given
in Figure 5. Then, we applied some random transformation
with condition number between one to two. For measuring

First, we evaluated the performance of our randomized ICA
approach (RICA) on two mixtures of independent sources
drawn in independently identically distributed fashion from
the pdfs in 5. We repeated the experiments for 250 and 1000
samples, and the Amari distance for various ICA algorithms
are summerized in Tables I and II, respectively. As expected,
the our results are comparable to those of KICA [1], while
our algorithm is strictly linear in the sample size.
We compared the robustness of our contrast function to
outlier samples. The evaluation was done by choosing at
random a certain number of points in the dataset and adding 5
or −5 at random with probability 0.5. The Amari distance
between the estimated matrix and the true one for several
algorithms is demonstrated in Figure 6. We repeated the
experiments 1000 times and averaged the results. The most
robust algorithms are KICA [1], while RICA algorithms are
still more robust then the others.
Since our framework imitates the kernel ICA methods in
the random feature space, we evaluated the accuracy and the
runtime of the methods in unmixing real audio signal. The
results are given in Table III. With comparable accuracy, our
methods run more than ten times faster.

TABLE II
T HE A MARI ERRORS ( MULTIPLIED BY 100) AND THE RUNTIME

Fig. 6. Robustness to outliers.
TABLE I
T HE A MARI ERRORS ( MULTIPLIED BY 100) OF STATE - OF - THE - ART ICA
ALGORITHMS EVALUATED ON A COUPLE OF MIXTURES OF INDEPENDENT
SOURCES DRAWN FROM THE DENOTED PDFS . I N THIS EXPERIMENT WE
USED 250 SAMPLES AND REPEATED THE EXPERIMENT 1000 TIMES . T HE
KGV, KCC, RCC, AND RGV ALGORITHMS WERE INITIALIZED WITH THE
SIMILAR MATRICES . E ACH ROW REPRESENT THE PDFS FROM WHICH THE
SAMPLES WERE DRAWN . T HE MEAN ROW REPRESENT THE AVERAGE
PERFORMANCE OF EACH METHOD AND RAND ROW DENOTES THE
PERFORMANCE OF THE ALGORITHMS FOR A MIXTURE OF A PAIRS OF
SOURCES RANDOMLY SELECTED FROM THE PDFS . W E REPEATED THIS
EXPERIMENT 1000 TIMES .

pdfs
a
b
c
d
e
f
g
h
i
j
k
l
m
n
o
p
q
r
mean
rand

F-ica
8.1
12.5
4.9
12.9
10.6
7.4
3.6
12.4
20.9
16.3
13.2
21.9
8.4
12.9
9.8
9.1
33.4
13.0
12.8
10.5

Jade
7.2
10.1
3.6
11.2
8.6
5.2
2.8
8.6
16.5
14.2
9.7
18.0
5.9
9.7
6.8
6.4
31.4
9.3
10.3
9.0

KCC
9.6
12.2
4.9
15.4
3.8
3.9
3.0
17.6
33.7
3.4
9.3
18.1
4.3
8.4
15.5
4.9
13.0
13.5
10.8
8.0

RCC
8.8
10.7
4.5
13.6
3.7
4.2
2.9
14.4
28.9
3.2
8.1
14.4
11.1
11.4
14.5
6.0
16.5
11.7
10.5
8.7

KGV
7.3
9.6
3.3
12.8
2.9
3.2
2.7
14.4
31.1
2.9
7.0
15.5
3.2
4.7
10.9
3.4
8.5
9.7
8.5
5.9

RGV
6.9
8.7
3.6
12.0
3.1
3.5
2.7
12.2
29.0
2.9
6.5
13.5
7.2
7.5
10.8
4.7
11.9
9.2
8.7
6.8

V. C ONCLUSIONS
We proposed two novel pseudo-contrast functions for solving the ICA problem. The functions are evaluated using randomized Fourier features and thus can be computed in linear
time. As the number of feature grows, the proposed functions
converge to the renowned kernel generalized variance the
kernel canonical correlation which require a computational
effort that is cubic in the sample size. The accuracy of the
proposed ICA methods is comparable to the state-of-the-art
but runs over ten times faster. The proposed functions could

pdfs
a
b
c
d
e
f
g
h
i
j
k
l
m
n
o
p
q
r
mean
rand

F-ica
4.2
5.9
2.2
6.8
5.2
3.7
1.7
5.4
9.0
5.8
6.3
9.4
3.7
5.3
4.2
4.0
16.4
5.7
5.8
5.8

Jade
3.6
4.7
1.6
5.3
4.0
2.6
1.3
3.9
6.6
4.2
4.4
6.7
2.6
3.7
3.1
2.7
12.4
4.0
4.3
4.6

KCCA
5.3
5.4
2.0
8.4
1.6
1.7
1.4
6.5
12.9
1.5
4.3
7.0
1.7
2.8
5.2
2.0
3.8
5.3
4.4
3.6

RCC
4.9
5.3
1.7
7.4
1.6
1.9
1.2
6.0
12.1
1.3
3.6
6.6
3.1
3.2
4.8
2.4
4.5
4.5
4.2
3.7

KGV
3.0
3.0
1.4
5.3
1.2
1.3
1.2
4.4
10.6
1.3
2.6
4.9
1.3
1.8
3.3
1.4
2.1
3.2
3.0
2.5

RGV
3.5
3.9
1.4
5.9
1.3
1.4
1.1
4.4
10.0
1.2
2.7
4.9
2.0
2.3
3.4
1.7
3.0
3.3
3.2
2.8

TABLE III
T HE A MARI ERRORS ( MULTIPLIED BY 100) AND THE RUNTIME ANALYSIS
IN SEPARATING A PAIR OF INDEPENDENT AUDIO SIGNALS FROM TWO
RECORDED MIXTURES .

Method
KCC
RCC
KGV
RGV

# repl
100
100
100
100

Amari Distance
3.2
3.3
1.2
1.3

Runtime (seconds)
337.7
28.4
258.0
23.6

also be evaluated using Nystrom extension for kernel matrices.
R EFERENCES

[1] F. R. Bach and M. I. Jordan, “Kernel independent component analysis,”
J. Mach. Learn. Res., vol. 3, pp. 1–48, Mar. 2003. [Online]. Available:
http://dx.doi.org/10.1162/153244303768966085
[2] A. Rahimi and B. Recht, “Weighted sums of random kitchen
sinks: Replacing minimization with randomization in learning,” in
Advances in Neural Information Processing Systems 21, D. Koller,
D. Schuurmans, Y. Bengio, and L. Bottou, Eds. Curran Associates, Inc.,
2009, pp. 1313–1320. [Online]. Available: http://papers.nips.cc/paper/
3495-weighted-sums-of-random-kitchen-sinks-replacing-minimization-with-randomizat
pdf
[3] D. Lopez-Paz, S. Sra, A. Smola, Z. Ghahramani, and B. Schölkopf, “Randomized nonlinear component analysis,” arXiv preprint arXiv:1402.0119,
2014.
[4] H. Hotelling, “Relations between two sets of variates,” Biometrika,
vol. 28, no. 3/4, pp. 321–377, 1936.
[5] S.-i. Amari, A. Cichocki, H. H. Yang et al., “A new learning algorithm
for blind signal separation,” Advances in neural information processing
systems, pp. 757–763, 1996.

