CN109583563A - Variation expectation maximization routing algorithm based on capsule network - Google Patents
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Abstract
The invention discloses a kind of variation expectation maximization routing algorithm based on capsule network, the pose matrix of rudimentary capsule is considered as the data point of GMM, the pose matrix of advanced capsule is considered as Gaussian Profile, data point cluster for Gaussian Profile one by one and is calculated into its distribution parameter by VBEM routing algorithm, it is grouped rudimentary capsule to form a high level capsule at runtime, is then updated according to Gaussian Distribution Parameters and calculate activation value a.The present invention uses VBEM algorithm in capsule network, compared with EM algorithm, this method hardly needs additional calculation amount, and it solves the calculating problem in maximum likelihood method, inferred based on variation and each factoring is optimized to complete whole optimization process, both available approximate solution, it again can be to avoid the singularity generated when Gaussian component " degeneration " is to a specific data point, but also hidden variable classification number k can be determined in the algorithm automatically, and over-fitting is also avoided when k is larger.
Description
Technical field
The present invention relates to variation infer and Bayes field, and in particular to it is a kind of based on capsule network variation expectation most
Bigization routing algorithm is used as and clusters rudimentary capsule for advanced capsule.
Background technique
Capsule network is a kind of new neural network model, and capsule is a concept of space, can be vector, is also possible to square
The concrete form of battle array scalar quantity, capsule can determine that capsule network is in Handwritten Digit Recognition field according to the feature of input data
Only need 3 layers of hidden layer that can achieve the effect that deep neural network, it is even better.
Expectation maximization (EM) algorithm is that the one kind for the maximum likelihood solution for finding the probabilistic model with latent variable is general
Method, gauss hybrid models (GMM) refer to the linear combination of multiple gauss of distribution function, are widely used in data mining, machine
In device study and statistical analysis.In numerous applications, parameter is determined by maximum likelihood method, it will usually use EM algorithm.However,
Maximum likelihood method has some huge limitations, and it is general about latent variable posteriority to need to calculate partial data log-likelihood function
The expectation of rate distribution.For many models in practical application, calculates Posterior probability distribution or calculate about after this
It is infeasible to test being contemplated to be for probability distribution, and in maximum likelihood method, when a Gaussian component " degeneration " has to one
When the data point of body, singularity can be generated, and this singularity is not present in bayes method.
The decomposition that the method that variation is inferred is based on true Posterior probability distribution is approximate, carries out most to each factoring
Optimization is to complete whole optimization process, using the method for variation Bayes, not only available approximate solution, but also can be to avoid working as
The singularity generated when Gaussian component " degeneration " is to a specific data point, the present invention are calculated in capsule network using VBEM
Method, compared with EM algorithm, this method hardly needs additional calculation amount, and it solves the master in maximum likelihood method
It wants difficult, but also hidden variable classification number k can be determined in the algorithm automatically, and also avoided intending when k is larger
It closes.VBEM algorithm is a two stage iteration optimization algorithms, is generally divided into VBE, VBM step, and VBE step is calculated according to parameter current
Posterior distrbutionp, VBM step is according to optimal solution form undated parameter, until convergence.
Summary of the invention
For solve existing capsule network Road by algorithm solve part practical problem can not computational problem, the present invention discloses
A kind of variation expectation maximization routing algorithm based on capsule network, variation expectation maximization routing algorithm purpose are glue
Capsule is grouped the relationship to form a part and entirety: the pose matrix of rudimentary capsule is considered as the data point of GMM, advanced capsule
Pose matrix is considered as Gaussian Profile, and data point cluster for Gaussian Profile one by one and is calculated its distribution ginseng by VBEM routing algorithm
Number, i.e., be grouped rudimentary capsule to form a high level capsule at runtime, is then updated and is calculated according to Gaussian Distribution Parameters
Activation value a.
Technical solution: to achieve the above object, the technical solution adopted by the present invention are as follows:
Variation expectation maximization routing algorithm based on capsule network uses VBEM routing algorithm meter in capsule network layer
Calculate the output attitude matrix and activation value of capsule.VBEM routing algorithm uses the Gauss model with 16 μ and 16 σ to father's capsule
Attitude matrix modeling, each μ indicates an element of attitude matrix, and σ is used for the calculating of activation value a, in VBEM routing algorithm
Two stages in constantly iteration until convergence, calculate the input pose matrix and activation value a of father's capsule；
S01 inputs current pose matrix and activation value, using the pose matrix of each capsule as data point X={ x1,
x2...xN, N is the number of capsule, using pose matrix classification as hidden variable Z={ z1,z2...zK, k be capsule classification, k by
VBEM algorithm automatically determines；
S02 infers the optimal of the optimal solution and posterior probability p (z | x) for finding out the prior distribution of each parameter θ using variation
Solve q*(z), θ includes mixed coefficint π={ π1π2...πk, mean μ={ μ of GMM1μ2...μk, covariance Λ={ Λ of GMM1
Λ2...Λk}；
S03 defines each variable and initializes, wherein being attached to Gaussian Profile initial value μ for pose matrix element is corresponding0,
It forces covariance Λ being set as diagonal matrix, then variances sigma2Take covariance matrix diagonal line；
S04 carries out VBM step, updates each variable；
S05 carries out VBE step, updates the r for representing posterior probabilitynk；
S06, iteration n times or until iteration convergence obtain final activation value a, and the mixed Gaussian mould that output x is obeyed
The mean value of type, conversion obtain father's capsule pose matrix.
Above-mentioned steps S02 is specifically included:
Data point x belongs to the probability distribution of kth class:
Wherein πkFor k-th of element of mixed coefficint π, μk,ΛkFor k-th of element of μ, Λ；
πkIndicate zk=1 probability, it may be assumed that p (zk=1)=πk, when k-th of element of z is 1, other elements are 0, institute
It is indicated with the marginal probability distribution of z are as follows:
By the optimal solution q of the prior probability q (π) of mixed coefficint π*(π) regards the distribution of Di Li Cray as, it may be assumed that
q*(π)=Dir (π | α), (3)
Wherein α is Di Li Cray coefficient；
Enable the optimal solution q of the mean μ of Gaussian Profile and the prior probability q (μ, Λ) of covariance Λ*(μ, Λ) obeys independent
Gauss-Wishart distribution, it may be assumed that
Wherein mk、βk、ωk、νkFor Gauss-Wishart distribution parameter；
The optimal solution q for directly writing out posterior probability p (z | x) is inferred according to variation*(z) logarithmic function:
Wherein:
lnρnzIt is lnq*(z) one represents the factor, and E [*] is to ask * expectation, and D is the dimension of data point x；
Index is gone to formula (5) two sides, obtains proportional relation are as follows:
It is required that probability be it is normalized, have:
Wherein:
rnkIt is ρnkNormalized form, rnkIt is non-negative, and
The above-mentioned steps S03 course of work includes the following steps: definition observation data for rnkBelow three statistics:
By rnkIt is considered as posterior probability, then NkIndicate kth class probability,Indicate the mean value for belonging to the x of kth class, SkIt indicates to belong to
In the covariance of the x of kth class, N is initializedk,Sk, the symbol lower right corner is designated as number 0 i.e. expression initial value in following formula, is
K represents its k-th of component；
(3-2) definition updates Di Li Cray parameter equation: αk=α0+Nk (13)
(3-3) definition updates Gauss-Wishart parameter equation: βk=β0+Nk (14)
vk=v0+NK+1 (17)
(3-4) definition updates mean value formula: μk=mk(18)
(3-5) definition updates formula of variance: σ2=diag (inv (Wk -1)) (19)
(3-6) definition updates capsule network cost function formulation: cost=βμ+lg(σ)∑rij (20)
Wherein βμFor network parameter, obtained by reversed Internet communication training；
(3-7) definition updates activation value a formula: aj=logstic (λ (βα-∑cost)) (21)
Wherein βαFor network parameter, obtained by reversed Internet communication training；
(3-8) initializes each variable defined above, wherein corresponding to be attached to Gaussian Profile initial by pose matrix element
Value μ0IfThen μ0=(μ1,μ2...μ15,μ16)T, the symbol lower right corner is designated as in following formula
Number 0 indicates initial value, all represents its k-th of component for k.
The above-mentioned steps S04 course of work includes the following steps:
(4-1) updates statistic Nk,Sk；
(4-3) updates Di Li Cray parameter alphak；
(4-4) updates Gauss-Wishart parameter: βk,mk,Wk -1,vk；
(4-5) updates mean μk, variances sigma2；
(4-6) updates capsule network cost function cost, updates activation value a.
The above-mentioned steps S05 course of work includes the following steps:
(5-1) updates ln ρ according to formula (6)nk, in which:
(5-2) updates r according to formula (9)nk。
Beneficial effects of the present invention: the present invention uses VBEM algorithm, compared with EM algorithm, this method in capsule network
Additional calculation amount is hardly needed, and it solves the calculating problem in maximum likelihood method, is inferred based on variation to every
A factoring is optimized to complete whole optimization process, not only available approximate solution, but also can be to avoid working as Gauss
The singularity generated when component " degeneration " is to a specific data point, but also hidden variable classification number k can be automatically in algorithm
Middle determination, and over-fitting is also avoided when k is larger.A kind of variation expectation maximization routing calculation based on capsule network
Method is a general-purpose algorithm.
Detailed description of the invention
Fig. 1 is VBEM algorithm routing procedure；
Scheme model structure in Fig. 2 embodiment of the present invention；
Fig. 3 is data structure diagram of the picture after every layer network in the embodiment of the present invention.
Specific embodiment
The invention will be further described with reference to the accompanying drawing and by specific embodiment, and following embodiment is descriptive
, it is not restrictive, this does not limit the scope of protection of the present invention.
In order to make technological means of the invention, creation characteristic, workflow, application method reach purpose and effect, and it is
It is easy to understand the evaluation method with reference to specific embodiments the present invention is further explained.
As shown in Figure 2 and Figure 3, use mnist data set as the training data of capsule network, initial data size is 28*
28, it with convolution kernel is 5x5 in common convolutional layer, step-length 2, have filling carries out convolution operation to picture, obtains 14*14*
32 input；
In initial capsule layer, 32 channels are converted into 32 primary capsules, each capsule packet using the convolution kernel of 1x1
Containing a 4x4 matrix and an activation value；
It is a convolution capsule layer 1, convolution kernel 3x3, step-length 2, the use of convolution capsule layer 1 after initial capsule layer
VBEM routing exports (pose matrix and activation value a) to calculate capsule；
Convolution capsule layer 2 is another convolution capsule layer, 3x3 convolution kernel, and step-length 1 is equally routed using VBEM to count
Calculate capsule output；
The output capsule of convolution capsule layer 2 is connected to classification capsule layer by 1x1 convolution kernel, each classification is by a glue
Capsule indicates (in MNIST, there is 10 classifications).
VBEM routing algorithm is realized:
As shown in Figure 1, VBEM routing algorithm uses the Gauss model with 16 μ and 16 σ to the attitude matrix of father's capsule
Modeling, each μ indicate an element of attitude matrix, and σ is used for the calculating of activation value a, in two stages of VBEM routing algorithm
In constantly iteration until convergence, calculate the input pose matrix and activation value a of father's capsule；
S01 inputs current pose matrix and activation value, using the pose matrix of each capsule as data point X={ x1,
x2...xN, N is the number of capsule, using pose matrix classification as hidden variable Z={ z1,z2...zK, k be capsule classification, k by
VBEM algorithm automatically determines；
S02 infers the optimal of the optimal solution and posterior probability p (z | x) for finding out the prior distribution of each parameter θ using variation
Solve q*(z), θ includes mixed coefficint π={ π1π2...πk, mean μ={ μ of GMM1μ2...μk, covariance Λ={ Λ of GMM1
Λ2...Λk}；
S03 defines each variable and initializes, wherein being attached to Gaussian Profile initial value μ for pose matrix element is corresponding0,
It forces covariance Λ being set as diagonal matrix, then variances sigma2Take covariance matrix diagonal line；
S04 carries out VBM step, updates each variable；
S05 carries out VBE step, updates the r for representing posterior probabilitynk；
S06, iteration n times or until iteration convergence obtain final activation value a, and the mixed Gaussian mould that output x is obeyed
The mean value of type, conversion obtain father's capsule pose matrix.
Above-mentioned steps S02 is specifically included:
Data point x belongs to the probability distribution of kth class:
Wherein πkFor k-th of element of mixed coefficint π, μk,ΛkFor k-th of element of μ, Λ；
πkIndicate zk=1 probability, it may be assumed that p (zk=1)=πk, when k-th of element of z is 1, other elements are 0, institute
It is indicated with the marginal probability distribution of z are as follows:
By the optimal solution q of the prior probability q (π) of mixed coefficint π*(π) regards the distribution of Di Li Cray as, it may be assumed that
q*(π)=Dir (π | α), (3)
Wherein α is Di Li Cray coefficient；
Enable the optimal solution q of the mean μ of Gaussian Profile and the prior probability q (μ, Λ) of covariance Λ*(μ, Λ) obeys independent
Gauss-Wishart distribution, it may be assumed that
Wherein mk、βk、ωk、νkFor Gauss-Wishart distribution parameter；
The optimal solution q for directly writing out posterior probability p (z | x) is inferred according to variation*(z) logarithmic function:
Wherein:
lnρnzIt is lnq*(z) one represents the factor, and E [*] is to ask * expectation, and D is the dimension of data point x；
Index is gone to formula (5) two sides, obtains proportional relation are as follows:
It is required that probability be it is normalized, have:
Wherein:
rnkIt is ρnkNormalized form, rnkIt is non-negative, and
The above-mentioned steps S03 course of work includes the following steps:
(3-1) definition observation data are for rnkBelow three statistics:
By rnkIt is considered as posterior probability, then NkIndicate kth class probability,Indicate the mean value for belonging to the x of kth class, SkIt indicates to belong to
In the covariance of the x of kth class, N is initializedk,Sk, the symbol lower right corner is designated as number 0 i.e. expression initial value in following formula, is
K represents its k-th of component；
(3-2) definition updates Di Li Cray parameter equation: αk=α0+Nk (13)
(3-3) definition updates Gauss-Wishart parameter equation: βk=β0+Nk (14)
vk=v0+NK+1 (17)
(3-4) definition updates mean value formula: μk=mk (18)
(3-5) definition updates formula of variance: σ2=diag (inv (Wk -1)) (19)
(3-6) definition updates capsule network cost function formulation: cost=βμ+lg(σ)∑rij (20)
Wherein βμFor network parameter, obtained by reversed Internet communication training；
(3-7) definition updates activation value a formula: aj=logstic (λ (βα-∑cost)) (21)
Wherein βαFor network parameter, obtained by reversed Internet communication training；
(3-8) initializes each variable defined above, wherein corresponding to be attached to Gaussian Profile initial by pose matrix element
Value μ0IfThen μ0=(μ1,μ2...μ15,μ16)T, the symbol lower right corner is designated as in following formula
Number 0 indicates initial value, all represents its k-th of component for k.
The above-mentioned steps S04 course of work includes the following steps:
(4-1) updates statistic Nk,Sk；
(4-3) updates Di Li Cray parameter alphak；
(4-4) updates Gauss-Wishart parameter: βk,mk,Wk -1,vk；
(4-5) updates mean μk, variances sigma2；
(4-6) updates capsule network cost function cost, updates activation value a.The above-mentioned steps S05 course of work includes as follows
Step:
(5-1) updates ln ρ according to formula (6)nk, in which:
(5-2) updates r according to formula (9)nk。
Claims (5)
1. a kind of variation expectation maximization routing algorithm based on capsule network, which is characterized in that
The pose matrix of rudimentary capsule is considered as the data point of GMM, the pose matrix of advanced capsule is considered as Gaussian Profile, by VBEM
Data point cluster for Gaussian Profile one by one and is calculated its distribution parameter by routing algorithm, i.e., is at runtime grouped rudimentary capsule
A high level capsule is formed, is then updated according to Gaussian Distribution Parameters and calculates activation value a；
S01 inputs current pose matrix and activation value, using the pose matrix of each capsule as data point X={ x1,x2...xN, N
For the number of capsule, using pose matrix classification as hidden variable Z={ z1,z2...zK, k be capsule classification, k by VBEM algorithm from
It is dynamic to determine；
S02 infers the optimal solution q of the optimal solution and posterior probability p (z | x) that find out the prior distribution of each parameter θ using variation*
(z), θ includes mixed coefficint π={ π1 π2 ... πk, mean μ={ μ of GMM1 μ2 ... μk, the covariance Λ of GMM=
{Λ1 Λ2 ... Λk}；
S03 defines each variable and initializes, wherein being attached to Gaussian Profile initial value μ for pose matrix element is corresponding0, forcing will
Covariance Λ is set as diagonal matrix, then variances sigma2Take covariance matrix diagonal line；
S04 carries out VBM step, updates each variable；
S05 carries out VBE step, updates the r for representing posterior probabilitynk；
S06, iteration n times or until iteration convergence obtain final activation value a, and the mixed Gauss model obeyed of output x
Mean value, conversion obtain father's capsule pose matrix.
2. a kind of variation expectation maximization routing algorithm based on capsule network according to claim 1, which is characterized in that
The step S02 is specifically included:
Data point x belongs to the probability distribution of kth class:
Wherein πkFor k-th of element of mixed coefficint π, μk,ΛkFor k-th of element of μ, Λ；
πkIndicate zk=1 probability, it may be assumed that p (zk=1)=πk, when k-th of element of z is 1, other elements are 0, so z
Marginal probability distribution indicates are as follows:
By the optimal solution q of the prior probability q (π) of mixed coefficint π*(π) regards the distribution of Di Li Cray as, it may be assumed that
q*(π)=Dir (π | α), (3)
Wherein α is Di Li Cray coefficient；
Enable the optimal solution q of the mean μ of Gaussian Profile and the prior probability q (μ, Λ) of covariance Λ*(μ, Λ) obeys independent height
This-Wishart distribution, it may be assumed that
Wherein mk、βk、ωk、νkFor Gauss-Wishart distribution parameter；
The optimal solution q for directly writing out posterior probability p (z | x) is inferred according to variation*(z) logarithmic function:
Wherein:
lnρnzIt is lnq*(z) one represents the factor, and E [*] is to ask * expectation, and D is the dimension of data point x；
Index is gone to formula (5) two sides, obtains proportional relation are as follows:
It is required that probability be it is normalized, have:
Wherein:
rnkIt is ρnkNormalized form, rnkIt is non-negative, and
3. a kind of variation expectation maximization routing algorithm based on capsule network according to claim 1, which is characterized in that
The step S03 course of work includes the following steps:
(3-1) definition observation data are for rnkFollowing three statistic:
By rnkIt is considered as posterior probability, then NkIndicate kth class probability,Indicate the mean value for belonging to the x of kth class, SkExpression belongs to kth
The covariance of the x of class initializes Nk,Sk, the symbol lower right corner, which is designated as number 0, in following formula indicates initial value, for k all generations
Its k-th of component of table；
(3-2) definition updates Di Li Cray parameter equation: αk=α0+Nk (13)
(3-3) definition updates Gauss-Wishart parameter equation: βk=β0+Nk (14)
vk=v0+NK+1 (17)
(3-4) definition updates mean value formula: μk=mk (18)
(3-5) definition updates formula of variance:
(3-6) definition updates capsule network cost function formulation: cost=βμ+lg(σ)∑rij (20)
Wherein βμFor network parameter, obtained by reversed Internet communication training；
(3-7) definition updates activation value a formula: aj=logstic (λ (βα-∑cost)) (21)
Wherein βαFor network parameter, obtained by reversed Internet communication training；
(3-8) initializes each variable defined above, wherein being attached to Gaussian Profile initial value μ for pose matrix element is corresponding0,
IfThen μ0=(μ1,μ2...μ15,μ16)T, the symbol lower right corner is designated as number in following formula
0 indicates initial value, all represents its k-th of component for k.
4. a kind of variation expectation maximization routing algorithm based on capsule network according to claim 1, which is characterized in that
The step S04 course of work includes the following steps:
(4-1) updates statistic Nk,Sk；
(4-3) updates Di Li Cray parameter alphak；
(4-4) updates Gauss-Wishart parameter: βk,mk,vk；
(4-5) updates mean μk, variances sigma2；
(4-6) updates capsule network cost function cost, updates activation value a.
5. a kind of variation expectation maximization routing algorithm based on capsule network according to claim 1, which is characterized in that
The step S05 course of work includes the following steps:
(5-1) updates ln ρ according to formula (6)nk, in which:
(5-2) updates r according to formula (9)nk。
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