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rli
pbdlib-matlab
Commits
1d1a390a
Commit
1d1a390a
authored
2 years ago
by
Sylvain CALINON
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TTGO example added
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1d1a390a
function
demo_tensor_TTGO01
% Global optimization with tensor trains (TTGO), consisting of:
% - Encoding the cost as a distribution with a low-rank factorization using TT-cross algorithm
% - (Conditional) sampling from the tensor train decomposition of the distribution
% (see https://sites.google.com/view/ttgo for details about the approach and for more elaborated codes in Python)
%
% If this code is useful for your research, please cite the related publication:
% @article{Shetty22,
% author={Shetty, S. and Lembono, T. and L\"ow, T. and Calinon, S.},
% title={Tensor Trains for Global Optimization Problems in Robotics},
% journal={arXiv:2206.05077},
% year={2022}
% }
%
% Copyright (c) 2022 Idiap Research Institute, https://idiap.ch/
% Written by Sylvain Calinon, https://calinon.ch/
%
% This file is part of PbDlib, http://www.idiap.ch/software/pbdlib/
%
% PbDlib is free software: you can redistribute it and/or modify
% it under the terms of the GNU General Public License version 3 as
% published by the Free Software Foundation.
%
% PbDlib is distributed in the hope that it will be useful,
% but WITHOUT ANY WARRANTY; without even the implied warranty of
% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
% GNU General Public License for more details.
%
% You should have received a copy of the GNU General Public License
% along with PbDlib. If not, see <http://www.gnu.org/licenses/>.
%% Parameters and data
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
r
=
3
;
%Rank for tensor decomposition
rTT
=
[
1
,
r
,
r
];
%TT-rank vector
n_max
=
500
;
%Maximum number of iterations
nbStates
=
3
;
%Number of Gaussians
Mu
=
[
1
2
2
;
3
6
2
;
4
1
3
];
%Means
sigma
=
[
1
,
1
,
1
]
*
1E0
;
%Variances
%Generating distribution as a GMM
nbVar
=
[
5
,
6
,
4
];
x
=
zeros
(
nbVar
);
for
i
=
1
:
nbVar
(
1
)
for
j
=
1
:
nbVar
(
2
)
for
k
=
1
:
nbVar
(
3
)
for
l
=
1
:
nbStates
eTmp
=
[
i
;
j
;
k
]
-
Mu
(:,
l
);
x
(
i
,
j
,
k
)
=
x
(
i
,
j
,
k
)
+
exp
(
-
eTmp
'*
eTmp
/
sigma
(
l
))
/
nbStates
;
end
end
end
end
x
=
x
+
rand
(
nbVar
)
*
1E-5
;
%% TT-cross approximation of the distribution
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
iv
{
1
}
=
randi
(
prod
(
nbVar
(
2
:
end
)),
[
r
,
1
]);
%Initial column list
iv
{
2
}
=
randi
(
prod
(
nbVar
(
3
:
end
)),
[
r
,
1
]);
%Initial column list
Pmat
{
1
}
=
reshape
(
x
,
[
nbVar
(
1
),
prod
(
nbVar
(
2
:
end
))]);
for
n
=
1
:
n_max
%Forward pass (select rows)
for
k
=
1
:
length
(
nbVar
)
-
1
U
=
qr
(
Pmat
{
k
}(:,
iv
{
k
}));
iu
{
k
}
=
maxvol
(
U
(:,
1
:
r
));
%Select rows
Pmat
{
k
+
1
}
=
reshape
(
Pmat
{
k
}(
iu
{
k
},:),
[
nbVar
(
k
+
1
)
*
rTT
(
k
+
1
),
prod
(
nbVar
(
k
+
2
:
end
))]);
end
%Backward pass (select columns)
for
k
=
length
(
nbVar
)
-
1
:
-
1
:
1
V
=
qr
(
Pmat
{
k
}(
iu
{
k
},:)
'
);
iv
{
k
}
=
maxvol
(
V
(:,
1
:
r
));
%Select columns
end
end
%disp(['TT-cross converged in ' num2str(n) ' iterations.']);
%Reconstruct TT-cores
for
k
=
1
:
length
(
nbVar
)
-
1
P
{
k
}
=
reshape
(
Pmat
{
k
}(:,
iv
{
k
})
/
Pmat
{
k
}(
iu
{
k
},
iv
{
k
}),
[
rTT
(
k
)
nbVar
(
k
)
rTT
(
k
+
1
)]);
end
P
{
length
(
nbVar
)}
=
reshape
(
Pmat
{
end
}(:,
1
),
[
rTT
(
end
)
nbVar
(
end
)]);
%Reconstruct the tensor from the TT-cores
x_est
=
zeros
(
nbVar
);
for
i
=
1
:
nbVar
(
1
)
for
j
=
1
:
nbVar
(
2
)
for
k
=
1
:
nbVar
(
3
)
x_est
(
i
,
j
,
k
)
=
reshape
(
P
{
1
}(
1
,
i
,:),
[
1
,
size
(
P
{
1
},
3
)])
*
reshape
(
P
{
2
}(:,
j
,:),
[
size
(
P
{
2
},
1
),
size
(
P
{
2
},
3
)])
*
reshape
(
P
{
3
}(:,
k
,
1
),
[
size
(
P
{
3
},
1
),
1
]);
end
end
end
%% TT sampling
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
alpha
=
.
2
;
%Prioritized sampling factor
N
=
500
;
%Number of sampled points
d
=
length
(
nbVar
);
%Order of tensor
Pi_hat
{
d
}
=
1
;
for
k
=
d
-
1
:
-
1
:
1
Pi_hat
{
k
}
=
squeeze
(
sum
(
P
{
k
+
1
},
2
))
*
Pi_hat
{
k
+
1
};
end
phi
{
1
}
=
ones
(
N
,
1
);
for
k
=
1
:
d
Pi
=
[];
for
xk
=
1
:
size
(
P
{
k
},
2
)
Pi
(:,
xk
)
=
reshape
(
P
{
k
}(:,
xk
,:),
[
size
(
P
{
k
},
1
),
size
(
P
{
k
},
3
)])
*
Pi_hat
{
k
};
end
for
l
=
1
:
N
p
=
abs
(
phi
{
k
}(
l
,:)
*
Pi
)
'
;
p
=
p
/
max
(
p
);
p
=
p
.^
(
1
/(
1
-
alpha
+
1E-9
));
%Prioritized sampling
p
=
p
/
sum
(
p
);
id
(
k
,
l
)
=
find
(
rand
<
cumsum
(
p
),
1
,
'first'
);
%Sampling from multinomial distribution (indices)
phi
{
k
+
1
}(
l
,:)
=
phi
{
k
}(
l
,:)
*
reshape
(
P
{
k
}(:,
id
(
k
,
l
),:),
[
size
(
P
{
k
},
1
),
size
(
P
{
k
},
3
)]);
end
end
%Sampled data (from indices)
x_gen
=
zeros
(
nbVar
);
for
l
=
1
:
N
x_gen
(
id
(
1
,
l
),
id
(
2
,
l
),
id
(
3
,
l
))
=
x_gen
(
id
(
1
,
l
),
id
(
2
,
l
),
id
(
3
,
l
))
+
1
;
end
x_gen
=
x_gen
*
max
(
x
(:))
/
max
(
x_gen
(:));
%max(x(:)) is given instead of 1 for color plots on the same scale
%% Plots with flattened 3D tensors
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
h
=
figure
(
'color'
,[
1
1
1
],
'position'
,[
10
,
10
,
1200
,
1400
]);
hold
on
;
axis
off
;
colormap
(
flipud
(
gray
));
%Original tensor
mplot
(
x
(:,:,
1
),
[
-
7
,
0
]);
mplot
(
x
(:,:,
2
),
[
-
7
,
7
]);
mplot
(
x
(:,:,
3
),
[
-
7
,
14
]);
mplot
(
x
(:,:,
4
),
[
-
7
,
21
]);
text
(
-
7
,
-
3
,
'Original tensor'
,
'horizontalalignment'
,
'center'
,
'fontsize'
,
30
);
%Estimated tensor
mplot
(
x_est
(:,:,
1
),
[
0
,
0
]);
mplot
(
x_est
(:,:,
2
),
[
0
,
7
]);
mplot
(
x_est
(:,:,
3
),
[
0
,
14
]);
mplot
(
x_est
(:,:,
4
),
[
0
,
21
]);
text
(
0
,
-
3
,
'Estimated tensor'
,
'horizontalalignment'
,
'center'
,
'fontsize'
,
30
);
%Tensor data samples
mplot
(
x_gen
(:,:,
1
),
[
7
,
0
]);
mplot
(
x_gen
(:,:,
2
),
[
7
,
7
]);
mplot
(
x_gen
(:,:,
3
),
[
7
,
14
]);
mplot
(
x_gen
(:,:,
4
),
[
7
,
21
]);
text
(
7
,
-
3
,
[
'Samples (N='
num2str
(
N
)
')'
],
'horizontalalignment'
,
'center'
,
'fontsize'
,
30
);
axis
ij
;
axis
equal
;
axis
tight
;
%print('-dpng','graphs/TTsampling01.png'); %'-r300',
waitfor
(
h
);
end
%% Maximal volume submatrix in an tall matrix based on LU decomposition
%% (returns rows indices that contain maximal volume submatrix)
function
id
=
maxvol
(
A
)
nbMaxIter
=
100
;
[
n
,
r
]
=
size
(
A
);
%Initialization
[
~
,
~
,
p
]
=
lu
(
A
,
'vector'
);
id
=
p
(
1
:
r
);
B
=
A
/
A
(
id
,:);
%Iterative algorithm
for
i
=
1
:
nbMaxIter
[
mx0
,
id2
]
=
max
(
abs
(
B
(:)));
[
i0
,
j0
]
=
ind2sub
([
n
,
r
],
id2
);
if
(
mx0
<=
1
+
5e-2
)
id
=
sort
(
id
);
break
;
end
k
=
id
(
j0
);
%This is the current row that we are using
B
=
B
+
B
(:,
j0
)
*
(
B
(
k
,:)
-
B
(
i0
,:))
/
B
(
i0
,
j0
);
id
(
j0
)
=
i0
;
end
end
%% Matrix plot function
function
h
=
mplot
(
c
,
pos
,
alpha
)
if
nargin
<
3
alpha
=
1
;
end
if
nargin
<
2
pos
=
[
0
,
0
];
end
pos
=
pos
-
[
size
(
c
,
2
),
size
(
c
,
1
)]
*
0.5
+
0.5
;
imagesc
([
0
,
size
(
c
,
2
)
-
1
]
+
pos
(
1
),
[
0
,
size
(
c
,
1
)
-
1
]
+
pos
(
2
),
c
);
rgx
=
[
0
:
size
(
c
,
2
)]
-
0.5
+
pos
(
1
);
rgy
=
[
0
:
size
(
c
,
1
)]
-
0.5
+
pos
(
2
);
line
(
repmat
(
rgx
,
2
,
1
),
repmat
([
rgy
(
1
);
rgy
(
end
)],
1
,
length
(
rgx
)),
'color'
,
[
0
0
0
]);
% vertical
line
(
repmat
([
rgx
(
1
);
rgx
(
end
)],
1
,
length
(
rgy
)),
repmat
(
rgy
,
2
,
1
),
'color'
,
[
0
0
0
]);
% horizontal
end
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