Device and method for training a variational autoencoder
A computer-implemented method for training a machine learning system. The training includes: determining, by an encoder of the machine learning system and based on a training input signal, a first intermediate representation characterizing a mean of a latent distribution of a latent space and a second intermediate representation characterizing a variance and/or covariance of the latent distribution; determining, based on the first intermediate representation and the second intermediate representation, a plurality of sigma points with respect to the latent distribution; determining an output signal, wherein the output signal is determined by providing a randomly sampled sigma point of the plurality of sigma points to a decoder of the machine learning system; adapting the machine learning system based on a loss value, wherein the loss value characterizes a difference between the training input signal and the output signal.
1 . A computer-implemented method for training a machine learning system, wherein the machine learning system is configured for anomaly detection and/or sampling a trajectory for a traffic participant and/or sampling of sensor signals and/or for determining a value characterizing a likelihood of an input signal with respect to a training dataset, wherein the training comprises the following steps:
determining, by an encoder of the machine learning system and based on a training input signal, a first intermediate representation characterizing a mean of a latent distribution of a latent space, and a second intermediate representation characterizing a variance and/or covariance of the latent distribution;
determining, based on the first intermediate representation and the second intermediate representation, a plurality of sigma points with respect to the latent distribution;
determining an output signal, wherein the output signal is determined by providing a randomly sampled sigma point of the plurality of sigma points to a decoder of the machine learning system; and
adapting the machine learning system based on a loss value, wherein the loss value characterizes a difference between the training input signal and the output signal,
wherein the sigma points in the plurality of sigma points are mean-centered symmetric points, comprising the mean characterized by the first intermediate representation;
wherein the plurality of sigma points is determined according to the formulae:
χ 0 =μ,
χ i =μ+√{square root over ((κ+ n )Σ)},
χ i+n =μ−√{square root over ((κ+ n )Σ)},
wherein κ>−n is a predefined real constant, n is a dimensionality of the latent space, μ is the mean, and Σ is the variance and/or covariance; and
wherein the loss value is determined based on a loss function, wherein the loss function is characterized by the formulae:
ℒ
U
A
E
=
E
x
i
∼
p
d
a
t
a
[
ℒ
R
E
C
+
β
·
ℒ
K
L
]
,
ℒ
R
E
C
=
x
i
-
D
(
z
)
2
2
,
z
∼
{
χ
i
(
μ
,
∑
)
}
i
=
0
2
n
,
ℒ
K
L
=
μ
2
2
+
tr
(
∑
)
-
n
-
log
det
∑
,
wherein x i is the training input signal, p data is an empirical distribution including a training dataset, D is the decoder of the machine learning system, and z is a randomly sampled sigma point of the plurality of sigma points
{
χ
i
(
μ
,
∑
)
}
i
=
0
2
n
.
2 . The method according to claim 1 , wherein the loss function further includes a regularization term characterized by the formula:
ℒ
R
E
G
=
λ
max
(
∑
)
∇
Z
D
(
z
)
2
2
,
wherein λ max is a largest eigen value of Σ and ∇ z D(z) is a gradient of the loss function with respect to z.
3 . The method according to claim 1 , wherein the sigma points in the plurality of sigma points are mean-centered symmetric points, including a mean characterized by the first intermediate representation.
4 . The method according to claim 1 , wherein the second intermediate representation characterizes a full covariant matrix of the latent distribution.
5 . The method according to claim 1 , wherein the training input signal is obtained based on a sensor.
6 . A computer-implemented method for determining whether an input signal is anomalous or normal, the method comprising the following steps:
obtaining a machine learning system that is configured for anomaly detection and that has been trained by:
determining, by an encoder of the machine learning system and based on a training input signal, a first intermediate representation characterizing a mean of a latent distribution of a latent space, and a second intermediate representation characterizing a variance and/or covariance of the latent distribution,
determining, based on the first intermediate representation and the second intermediate representation, a plurality of sigma points with respect to the latent distribution,
determining an output signal, wherein the output signal is determined by providing a randomly sampled sigma point of the plurality of sigma points to a decoder of the machine learning system, and
adapting the machine learning system based on a loss value, wherein the loss value characterizes a difference between the training input signal and the output signal;
providing the input signal to the encoder of the machine learning system to determine the first intermediate representation;
determining an output signal by providing the first intermediate representation as input to the decoder of the machine learning system;
determining the input signal as anomalous based on a difference between the output signal and the input signal exceeding a predefined threshold and otherwise determining the input signal as normal,
wherein the sigma points in the plurality of sigma points are mean-centered symmetric points, comprising the mean characterized by the first intermediate representation;
wherein the plurality of sigma points is determined according to the formulae:
χ 0 =μ,
χ i =μ+√{square root over ((κ+ n )Σ)},
χ i+n =μ−√{square root over ((κ+ n )Σ)},
wherein κ>−n is a predefined real constant, n is a dimensionality of the latent space, μ is the mean, and Σ is the variance and/or covariance; and
wherein the loss value is determined based on a loss function, wherein the loss function is characterized by the formulae:
ℒ
U
A
E
=
E
x
i
∼
p
d
a
t
a
[
ℒ
R
E
C
+
β
·
ℒ
K
L
]
,
ℒ
R
E
C
=
x
i
-
D
(
z
)
2
2
,
z
∼
{
χ
i
(
μ
,
∑
)
}
i
=
0
2
n
,
ℒ
K
L
=
μ
2
2
+
tr
(
∑
)
-
n
-
log
det
∑
,
wherein x i is the training input signal, p data is an empirical distribution, including a training dataset, D is the decoder of the machine learning system, and z is a randomly sampled sigma point of the plurality of sigma points
{
χ
i
(
μ
,
∑
)
}
i
=
0
2
n
.
7 . A computer-implemented method for sampling a trajectory of a traffic participant and/or a sampling sensor signal comprising the following steps:
obtaining a machine learning that has been trained and that is configured for trajectory sampling and/or sampling a sensor signal, the training including:
determining, by an encoder of the machine learning system and based on a training input signal, a first intermediate representation characterizing a mean of a latent distribution of a latent space, and a second intermediate representation characterizing a variance and/or covariance of the latent distribution,
determining, based on the first intermediate representation and the second intermediate representation, a plurality of sigma points with respect to the latent distribution,
determining an output signal, wherein the output signal is determined by providing a randomly sampled sigma point of the plurality of sigma points to a decoder of the machine learning system, and
adapting the machine learning system based on a loss value, wherein the loss value characterizes a difference between the training input signal and the output signal;
randomly drawing a value from the latent space characterized by the machine learning system; and
determining an output signal characterizing a trajectory by providing the randomly drawn value to the decoder of the machine learning system:
wherein the sigma points in the plurality of sigma points are mean-centered symmetric points, comprising the mean characterized by the first intermediate representation:
wherein the plurality of sigma points is determined according to the formulae:
χ 0 =μ,
χ i =μ+√{square root over ((κ+ n )Σ)},
χ i+n =μ−√{square root over ((κ+ n )Σ)},
wherein κ>−n is a predefined real constant, n is a dimensionality of the latent space, μ is the mean, and Σ is the variance and/or covariance; and
wherein the loss value is determined based on a loss function, wherein the loss function is characterized by the formulae:
ℒ
U
A
E
=
E
x
i
∼
p
d
a
t
a
[
ℒ
R
E
C
+
β
·
ℒ
K
L
]
,
ℒ
R
E
C
=
x
i
-
D
(
z
)
2
2
,
z
∼
{
χ
i
(
μ
,
∑
)
}
i
=
0
2
n
,
ℒ
K
L
=
μ
2
2
+
tr
(
∑
)
-
n
-
log
det
∑
,
wherein x i is the training input signal, p data is an empirical distribution including a training dataset, D is the decoder of the machine learning system, and z is a randomly sampled sigma point of the plurality of sigma points
{
χ
i
(
μ
,
∑
)
}
i
=
0
2
n
.
8 . A training system comprising at least one processor configured to train a machine learning system, wherein the machine learning system is configured for executed by the least one processor for anomaly detection and/or sampling a trajectory for a traffic participant and/or sampling of sensor signals and/or for determining a value characterizing a likelihood of an input signal with respect to a training dataset, wherein the training system is configured executed by the at least one processor to:
determine, using an encoder of the machine learning system and based on a training input signal, a first intermediate representation characterizing a mean of a latent distribution of a latent space, and a second intermediate representation characterizing a variance and/or covariance of the latent distribution;
determine, based on the first intermediate representation and the second intermediate representation, a plurality of sigma points with respect to the latent distribution;
determine an output signal, wherein the output signal is determined by providing a randomly sampled sigma point of the plurality of sigma points to a decoder of the machine learning system; and
adapt the machine learning system based on a loss value, wherein the loss value characterizes a difference between the training input signal and the output signal,
wherein the sigma points in the plurality of sigma points are mean-centered symmetric points, comprising the mean characterized by the first intermediate representation;
wherein the plurality of sigma points is determined according to the formulae:
χ 0 =μ,
χ i =μ+√{square root over ((κ+ n )Σ)},
χ i+n =μ−√{square root over ((κ+ n )Σ)},
wherein κ>−n is a predefined real constant, n is a dimensionality of the latent space, μ is the mean, and Σ is the variance and/or covariance; and
wherein the loss value is determined based on a loss function, wherein the loss function is characterized by the formulae:
ℒ
U
A
E
=
E
x
i
∼
p
d
a
t
a
[
ℒ
R
E
C
+
β
·
ℒ
K
L
]
,
ℒ
R
E
C
=
x
i
-
D
(
z
)
2
2
,
z
∼
{
χ
i
(
μ
,
∑
)
}
i
=
0
2
n
,
ℒ
K
L
=
μ
2
2
+
tr
(
∑
)
-
n
-
log
det
∑
,
wherein x i is the training input signal, p data is an empirical distribution including a training dataset, D is the decoder of the machine learning system, and z is a randomly sampled sigma point of the plurality of sigma points
{
χ
i
(
μ
,
∑
)
}
i
=
0
2
n
.
9 . A control system comprising at least one processor, which the at least one processor is configured to:
obtain a machine learning system that is configured for anomaly detection and that has been trained by:
determining, by an encoder of the machine learning system and based on a training input signal, a first intermediate representation characterizing a mean of a latent distribution of a latent space, and a second intermediate representation characterizing a variance and/or covariance of the latent distribution,
determining, based on the first intermediate representation and the second intermediate representation, a plurality of sigma points with respect to the latent distribution,
determining an output signal, wherein the output signal is determined by providing a randomly sampled sigma point of the plurality of sigma points to a decoder of the machine learning system, and
adapting the machine learning system based on a loss value, wherein the loss value characterizes a difference between the training input signal and the output signal;
provide an input signal to the encoder of the machine learning system to determine the first intermediate representation;
determine an output signal by providing the first intermediate representation as input to the decoder of the machine learning system;
determine the input signal as anomalous based on a difference between the output signal and the input signal exceeding a predefined threshold and otherwise determining the input signal as normal;
wherein the control system determines a control signal based on the output signal, wherein the control signal is configured to control an actuator and/or a display;
wherein the sigma points in the plurality of sigma points are mean-centered symmetric points, comprising the mean characterized by the first intermediate representation;
wherein the plurality of sigma points is determined according to the formulae:
χ 0 =μ,
χ i =μ+√{square root over ((κ+ n )Σ)},
χ i+n =μ−√{square root over ((κ+ n )Σ)},
wherein κ>−n is a predefined real constant, n is a dimensionality of the latent space, μ is the mean, and Σ is the variance and/or covariance; and
wherein the loss value is determined based on a loss function, wherein the loss function is characterized by the formulae:
ℒ
U
A
E
=
E
x
i
∼
p
d
a
t
a
[
ℒ
R
E
C
+
β
·
ℒ
K
L
]
,
ℒ
R
E
C
=
x
i
-
D
(
z
)
2
2
,
z
∼
{
χ
i
(
μ
,
∑
)
}
i
=
0
2
n
,
ℒ
K
L
=
μ
2
2
+
tr
(
∑
)
-
n
-
log
det
∑
,
wherein x i is the training input signal, p data is an empirical distribution including a training dataset, D is the decoder of the machine learning system, and z is a randomly sampled sigma point of the plurality of sigma points
{
χ
i
(
μ
,
∑
)
}
i
=
0
2
n
.
10 . A non-transitory machine-readable storage medium on which is stored a computer program for training a machine learning system, wherein the machine learning system is configured for anomaly detection and/or sampling a trajectory for a traffic participant and/or sampling of sensor signals and/or for determining a value characterizing a likelihood of an input signal with respect to a training dataset, wherein the computer program, when executed by a processor, causing the processor to train the machine learning system by performing the following steps:
determining, by an encoder of the machine learning system and based on a training input signal, a first intermediate representation characterizing a mean of a latent distribution of a latent space, and a second intermediate representation characterizing a variance and/or covariance of the latent distribution;
determining, based on the first intermediate representation and the second intermediate representation, a plurality of sigma points with respect to the latent distribution;
determining an output signal, wherein the output signal is determined by providing a randomly sampled sigma point of the plurality of sigma points to a decoder of the machine learning system; and
adapting the machine learning system based on a loss value, wherein the loss value characterizes a difference between the training input signal and the output signal,
wherein the sigma points in the plurality of sigma points are mean-centered symmetric points, comprising the mean characterized by the first intermediate representation;
wherein the plurality of sigma points is determined according to the formulae:
χ 0 =μ,
χ i =μ+√{square root over ((κ+ n )Σ)},
χ i+n =μ−√{square root over ((κ+ n )Σ)},
wherein κ>−n is a predefined real constant, n is a dimensionality of the latent space, μ is the mean, and Σ is the variance and/or covariance; and
wherein the loss value is determined based on a loss function, wherein the loss function is characterized by the formulae:
ℒ
U
A
E
=
E
x
i
∼
p
d
a
t
a
[
ℒ
R
E
C
+
β
·
ℒ
K
L
]
,
ℒ
R
E
C
=
x
i
-
D
(
z
)
2
2
,
z
∼
{
χ
i
(
μ
,
∑
)
}
i
=
0
2
n
,
ℒ
K
L
=
μ
2
2
+
tr
(
∑
)
-
n
-
log
det
∑
,
wherein x i is the training input signal, p data is an empirical distribution including a training dataset, D is the decoder of the machine learning system, and z is a randomly sampled sigma point of the plurality of sigma points
{
χ
i
(
μ
,
∑
)
}
i
=
0
2
n
.