CN107066566A - It is a kind of based on the network reasoning method for changing over time figure lasso trick - Google Patents

It is a kind of based on the network reasoning method for changing over time figure lasso trick Download PDF

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CN107066566A
CN107066566A CN201710216779.4A CN201710216779A CN107066566A CN 107066566 A CN107066566 A CN 107066566A CN 201710216779 A CN201710216779 A CN 201710216779A CN 107066566 A CN107066566 A CN 107066566A
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夏春秋
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Shenzhen Vision Technology Co Ltd
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Abstract

What is proposed in the present invention is a kind of based on the network reasoning method for changing over time figure lasso trick, and its main contents includes：Dynamic network, alternating direction Multiplier Algorithm (ADMM), Θ update, Z updates, its process is, figure lasso trick problem is expanded into dynamic network from static network first, then the time-varying figure lasso trick algorithm based on multiplier alternating direction (ADMM) is used by PROBLEM DECOMPOSITION, it is divided into each individually to update Θ, it is then parallel to solve；Z is updated and is divided into two parts, and parallelization and speed-up computation are carried out to each part, ADMM is updated.The present invention can estimate network simultaneously in itself and its structure changes with time with the time-varying figure lasso trick algorithm based on multiplier alternating direction (ADMM)；Meanwhile, it calculates easy, and accuracy is high, and scalability is strong, can efficiently solve problem.

Description

Network reasoning method based on time-varying graph lasso
Technical Field
The invention relates to the field of graph lassos, in particular to a time-varying graph lasso-based network reasoning method.
Background
Many important questions can be modeled as a system of interconnected entities, recording each entity's time-related observations or measurements, interpreting the temporal dynamics of these data by finding trends, detecting anomalies, understanding the relationships between different entities and how these relationships will develop over time, which is important to our insight into the question. We can solve many other problems by detecting anomalies at the intersection of time series analysis and network science, analyzing site trends, classifying events, predicting future behavior, etc. However, since a dynamic network is complex, it is difficult to simultaneously estimate the change of the network itself and the structure thereof over time, and the algorithm is complex, so that a novel algorithm and technology are required.
The invention provides a network reasoning method based on a time-varying graph lasso, which comprises the steps of firstly expanding the graph lasso problem from a static network to a dynamic network, then decomposing the problem by using a time-varying graph lasso algorithm based on an Alternating Direction (ADMM) of a multiplier, dividing theta into individual updates, and then solving the problems in parallel; the Z update is split into two parts and parallelized and accelerated computations are performed on each part, updating the ADMM. The invention uses the time-varying graph lasso algorithm based on the Alternative Direction (ADMM) of the multiplier, and can simultaneously estimate the change of the network and the structure thereof along with the time; meanwhile, the method is simple and convenient to calculate, high in accuracy and strong in expandability, and can effectively solve the problem.
Disclosure of Invention
Aiming at the problem of complex algorithm, the invention aims to provide a network reasoning method based on time-varying graph lasso, which comprises the steps of firstly expanding the graph lasso problem from a static network to a dynamic network, then decomposing the problem by using a time-varying graph lasso algorithm based on the Alternating Direction (ADMM) of a multiplier, dividing theta into individual updates, and then solving the problems in parallel; the Z update is split into two parts and parallelized and accelerated computations are performed on each part, updating the ADMM.
In order to solve the above problems, the present invention provides a network inference method based on a time-varying graph lasso, which mainly comprises:
a dynamic network;
(II) an alternating direction multiplier Algorithm (ADMM);
(III) updating;
and (IV) updating the Z.
The dynamic network allows ∑ (t) to change along with time by expanding the static network, and solves the problem of theta (theta)1,…,ΘT) At time t1,…,tTThe inverse covariance matrix of the inter-estimation,
here, |i(Θi)＝ni(logdetΘi-Tr(SiΘi)),β≥0，ψ(Θi-Θi-1) Is a convex function, minimized at ψ (0), which encourages Θt-1And ΘtThe similarity between them.
Wherein the alternating direction multiplier Algorithm (ADMM) is relative to the matrix A ∈ Rm×nAnd a real-valued function f (x), the near-end operator being defined as:
the near-end operator defines a trade-off of X between minimizing f and being close to A; decompose problem (1), introduce variable Z ═ Z0,Z1,Z2}＝{(Z1,0,…,ZT,0),(Z1,1,…,ZT-1,1),(Z2,2,…,ZT,2)}；
MinimizationIs subject to (Zi-1,1,Zi,2)＝(Θi-1,Θi)(i＝2,…,T)；
(a)
(b)
(c)
ADMM consists of the above updates, where k denotes the number of iterations.
Wherein, the updating of the theta, the step of the theta can be divided into each thetaiThen solved in parallel:
since thetaiIs suitable forAndthe near-end operator can be rewritten:
due to the fact thatIs symmetric, so there is an analytical solution:
wherein, QDQTIs thatThe characteristics of (1).
Wherein, the Z updating can be divided into two parts: z0It refers to the "Θ |" that enforces sparsity in the inverse covariance matrixod,1-a penalty term; (Z)1,Z2) ψ -penalty term representing minimum cross-time deviation; these two updates can be resolved simultaneously, and each part can be parallelized to accelerate the computation.
Further, Z is0Update, each Zi,0Can be used asNear-end operator of norm with known closed-form solution
Wherein,is an element intelligence-threshold function, i ≠ j ≠ p ≤ 1;
the (i, j) th element of this update is above.
Further, said (Z)1,Z2) Update, Zi-1,1And Zi,2To enhance Lagrangian bonding together and therefore must be combined moreNew; to obtain a closed form solution, define:
for each (Z)1,Z2) In order to resolve a single update, the update is,
the results are shown in the above formula.
Further, the column specifies (Z) of the sum1,Z2) Update, orderg＝ψ，C＝[-I I]D is 0 andwill (Z)1,Z2) Update conversion to
Wherein,due to the fact thatAndψ is the sum of column norms, so that E can be divided and simplified
[E]j＝(proxηψ(A))j＝proxηφ([A]j) (10)
Wherein phi isAndthe column norm of (d); will (Z) in the formula (4)i-1,1,Zi,2) The update is narrowed to find E in equation (11)jThe approximation operator of phi, defined by vectors, indicates that each vector has a closed form solution.
Further, the closed form solution, element intelligent productIn the penalty term, the number of the first time,the near-end operator of is The soft thresholds for the element intelligence product are as follows:
combined lassoIn a penalty term, used forThe near-end operator of the norm is a modular soft threshold,
for laplace penalty termNorm, the near-end operator of Laplace regularization is Re-editing into E in elemental formij＝(1+2η)-1(Aij)；
In the penalty term, the number of the first time,the near-end operator of the norm is
Wherein σ isThe solution of (1).
Further, the penalty term (Z) of the disturbing node1,Z2) Update to minimize the variable (Z)i-1,1,Zi,2) Is represented by (Y)1,Y2) (ii) a Then an additional variable V ═ W is introducedTIncreased LagrangianBecome into
Wherein,is a scaled bivariate, ρ is the same ADMM penalty parameter as the external ADMM; in the l-th iteration, three steps in the ADMM update are as follows:
(a)
(b)
(c)
whereinC＝[I -I I]，
Drawings
FIG. 1 is a system framework diagram of a time-varying graph lasso-based network inference method of the present invention.
FIG. 2 is a dynamic network of a time-varying graph lasso based network inference method of the present invention.
FIG. 3 is an alternative direction multiplier algorithm of the network inference method based on time-varying graph lasso according to the present invention.
Detailed Description
It should be noted that the embodiments and features of the embodiments in the present application can be combined with each other without conflict, and the present invention is further described in detail with reference to the drawings and specific embodiments.
FIG. 1 is a system framework diagram of a time-varying graph lasso-based network inference method of the present invention. The method mainly comprises a dynamic network, an alternating direction multiplier Algorithm (ADMM), theta updating and Z updating.
Theta update, the theta step can be split into each thetaiThen solved in parallel:
since thetaiIs suitable forAndthe near-end operator can be rewritten:
due to the fact thatIs symmetric, so there is an analytical solution:
wherein, QDQTIs thatThe characteristics of (1).
Z-update can be divided into two parts：Z0It refers to the "Θ |" that enforces sparsity in the inverse covariance matrixod,1-a penalty term; (Z)1,Z2) ψ -penalty term representing minimum cross-time deviation; these two updates can be resolved simultaneously, and each part can be parallelized to accelerate the computation.
Z0Update, each Zi,0Can be used asNear-end operator of norm with known closed-form solution
Wherein,is an element intelligence-threshold function, i ≠ j ≠ p ≤ 1;
the (i, j) th element of this update is above.
(Z1,Z2) Update, Zi-1,1And Zi,2To enhance lagrangian ties together, and therefore must be jointly updated; to obtain a closed form solution, define:
for each (Z)1,Z2) In order to resolve a single update, the update is,
the results are shown in the above formula.
Column normalized sum of (Z)1,Z2) Update, orderg＝ψ，C＝[-I I]D is 0 andwill (Z)1,Z2) Update conversion to
Wherein,due to the fact thatAndψ is the sum of column norms, so that E can be divided and simplified
[E]j＝(proxηψ(A))j＝proxηφ([A]j) (8)
Wherein phi isAndthe column norm of (d); will be (Z) in the formula (2)i-1,1,Zi,2) The update is narrowed to find E in equation (9)jThe approximation operator of phi, defined by vectors, indicates that each vector has a closed form solution.
Closed form solution, element intelligenceProduct ofIn the penalty term, the number of the first time,the near-end operator of isThe soft thresholds for the element intelligence product are as follows:
combined lassoIn a penalty term, used forThe near-end operator of the norm is a modular soft threshold,
for laplace penalty termNorm, the near-end operator of Laplace regularization is Re-editing into E in elemental formij＝(1+2η)-1(Aij)；
In the penalty term, the number of the first time,the near-end operator of the norm is
Wherein σ isThe solution of (1).
Perturbed node penalty term (Z)1,Z2) Update to minimize the variable (Z)i-1,1,Zi,2) Is represented by (Y)1,Y2) (ii) a Then an additional variable V ═ W is introducedTIncreased LagrangianBecome into
Wherein,is a scaled bivariate, ρ is the same ADMM penalty parameter as the external ADMM; in the l-th iteration, three steps in the ADMM update are as follows:
(a)
(b)
(c)
whereinC＝[I -I I]，
FIG. 2 is a dynamic network of the network inference method based on the time-varying graph lasso, ∑ (t) is allowed to vary with time by expanding a static network, and theta (theta) is solved1,…,ΘT) At time t1,…,tTThe inverse covariance matrix of the inter-estimation,
here, |i(Θi)＝ni(logdetΘi-Tr(SiΘi)),β≥0，ψ(Θi-Θi-1) Is a convex function, minimized at ψ (0), which encourages Θt-1And ΘtThe similarity between them.
FIG. 3 is an alternative direction multiplier algorithm for a time-varying graph lasso based network inference method of the present invention, relative to a matrix A ∈ Rm×nAnd a real-valued function f (x), the near-end operator being defined as:
the near-end operator defines a trade-off of X between minimizing f and being close to A; decomposing the problem (13), introducing the variable Z ═ Z0,Z1,Z2}＝{(Z1,0,…,ZT,0),(Z1,1,…,ZT-1,1),(Z2,2,…,ZT,2)}；
MinimizationIs subject to (Zi-1,1,Zi,2)＝(Θi-1,Θi)(i＝2,…,T)；
(a)
(b)
(c)
ADMM consists of the above updates, where k denotes the number of iterations.
It will be appreciated by persons skilled in the art that the invention is not limited to details of the foregoing embodiments and that the invention can be embodied in other specific forms without departing from the spirit or scope of the invention. In addition, various modifications and alterations of this invention may be made by those skilled in the art without departing from the spirit and scope of this invention, and such modifications and alterations should also be viewed as being within the scope of this invention. It is therefore intended that the following appended claims be interpreted as including preferred embodiments and all such alterations and modifications as fall within the scope of the invention.

Claims (10)

1. A network reasoning method based on a time-varying graph lasso is characterized by mainly comprising a dynamic network I; an alternating direction multiplier Algorithm (ADMM) (two); updating theta (III); z update (four).
2. Dynamic network (one) according to claim 1, characterized in that ∑ (t) is allowed to vary over time by extending the static network, and Θ (Θ) is solved1,…,ΘT) At time t1,…,tTInter-estimation inverse covariance matrix，
<mrow> <munder> <mrow> <mi>min</mi> <mi>i</mi> <mi>m</mi> <mi>i</mi> <mi>z</mi> <mi>e</mi> </mrow> <mrow> <mi>&amp;Theta;</mi> <mo>&amp;Element;</mo> <msubsup> <mi>S</mi> <mrow> <mo>+</mo> <mo>+</mo> </mrow> <mi>p</mi> </msubsup> </mrow> </munder> <munderover> <mi>&amp;Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>T</mi> </munderover> <mo>-</mo> <msub> <mi>l</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>&amp;Theta;</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mi>&amp;lambda;</mi> <mo>|</mo> <mo>|</mo> <mi>&amp;Theta;</mi> <mo>|</mo> <msub> <mo>|</mo> <mrow> <mi>o</mi> <mi>d</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>&amp;beta;</mi> <munderover> <mi>&amp;Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>2</mn> </mrow> <mi>T</mi> </munderover> <mi>&amp;psi;</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;Theta;</mi> <mi>i</mi> </msub> <mo>-</mo> <msub> <mi>&amp;Theta;</mi> <mrow> <mi>i</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow>
Here, |i(Θi)＝ni(logdetΘi-Tr(SiΘi)),β≥0，ψ(Θi-Θi-1) Is a convex function, minimized at ψ (0), which encourages Θt-1And ΘtThe similarity between them.
3. Based on the rightThe alternating direction multiplier Algorithm (ADMM) (II) of claim 1, wherein the matrix A ∈ R is a matrixm×nAnd a real-valued function f (x), the near-end operator being defined as:
<mrow> <msub> <mi>prox</mi> <mrow> <mi>&amp;eta;</mi> <mi>f</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>A</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>arg</mi> <munder> <mi>min</mi> <mrow> <mi>X</mi> <mo>&amp;Element;</mo> <msup> <mi>R</mi> <mrow> <mi>m</mi> <mo>&amp;times;</mo> <mi>n</mi> </mrow> </msup> </mrow> </munder> <mrow> <mo>(</mo> <mi>f</mi> <mo>(</mo> <mi>X</mi> <mo>)</mo> <mo>+</mo> <mn>1</mn> <mo>/</mo> <mo>(</mo> <mrow> <mn>2</mn> <mi>&amp;eta;</mi> </mrow> <mo>)</mo> <mo>|</mo> <mo>|</mo> <mi>X</mi> <mo>-</mo> <mi>A</mi> <mo>|</mo> <msubsup> <mo>|</mo> <mi>F</mi> <mn>2</mn> </msubsup> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow>
the near-end operator defines a trade-off of X between minimizing f and being close to A; decompose problem (1), introduce variable Z ═ Z0,Z1,Z2}＝{(Z1,0,…,ZT,0),(Z1,1,…,ZT-1,1),(Z2,2,…,ZT,2)}；
MinimizationSubject to Zi,0＝Θi,(i＝1,…,T)，(Zi-1,1,Zi,2)＝(Θi-1,Θi)(i＝2,…,T)；
(a)
(b)
(c)
ADMM consists of the above updates, where k denotes the number of iterations.
4.Θ update (III) based on claim 1, characterized in that the Θ step can be split into each ΘiThen solved in parallel:
<mrow> <mtable> <mtr> <mtd> <mrow> <msubsup> <mi>&amp;Theta;</mi> <mi>i</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>=</mo> <mi>arg</mi> <munder> <mi>min</mi> <mrow> <msub> <mi>&amp;Theta;</mi> <mi>i</mi> </msub> <mo>&amp;Element;</mo> <msubsup> <mi>S</mi> <mrow> <mo>+</mo> <mo>+</mo> </mrow> <mi>p</mi> </msubsup> </mrow> </munder> <mo>-</mo> <mi>log</mi> <mi> </mi> <mi>det</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;Theta;</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mi>T</mi> <mi>r</mi> <mrow> <mo>(</mo> <msub> <mi>S</mi> <mi>i</mi> </msub> <msub> <mi>&amp;Theta;</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>&amp;eta;</mi> </mrow> </mfrac> <mo>|</mo> <mo>|</mo> <mi>X</mi> <mo>-</mo> <msub> <mi>&amp;Theta;</mi> <mi>i</mi> </msub> <mo>|</mo> <msubsup> <mo>|</mo> <mi>F</mi> <mn>2</mn> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mi>arg</mi> <munder> <mi>min</mi> <mrow> <msub> <mi>&amp;Theta;</mi> <mi>i</mi> </msub> <mo>&amp;Element;</mo> <msubsup> <mi>S</mi> <mrow> <mo>+</mo> <mo>+</mo> </mrow> <mi>p</mi> </msubsup> </mrow> </munder> <mo>-</mo> <mi>log</mi> <mi> </mi> <mi>det</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;Theta;</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mi>T</mi> <mi>r</mi> <mrow> <mo>(</mo> <msub> <mi>S</mi> <mi>i</mi> </msub> <msub> <mi>&amp;Theta;</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>&amp;eta;</mi> </mrow> </mfrac> <mo>|</mo> <mo>|</mo> <msub> <mi>&amp;Theta;</mi> <mi>i</mi> </msub> <mo>-</mo> <mfrac> <mrow> <mi>A</mi> <mo>+</mo> <msup> <mi>A</mi> <mi>T</mi> </msup> </mrow> <mn>2</mn> </mfrac> <mo>|</mo> <msubsup> <mo>|</mo> <mi>F</mi> <mn>2</mn> </msubsup> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow>
since thetaiIs suitable forAndthe near-end operator can be rewritten:
<mrow> <msubsup> <mi>&amp;Theta;</mi> <mi>i</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>=</mo> <msub> <mi>prox</mi> <mrow> <mi>&amp;eta;</mi> <mrow> <mo>(</mo> <mo>-</mo> <mi>log</mi> <mi>det</mi> <mo>(</mo> <mo>&amp;CenterDot;</mo> <mo>)</mo> <mo>+</mo> <mi>T</mi> <mi>r</mi> <mo>(</mo> <mrow> <msub> <mi>S</mi> <mi>i</mi> </msub> <mo>&amp;CenterDot;</mo> </mrow> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </msub> <mrow> <mo>(</mo> <mfrac> <mrow> <mi>A</mi> <mo>+</mo> <msup> <mi>A</mi> <mi>T</mi> </msup> </mrow> <mn>2</mn> </mfrac> <mo>)</mo> </mrow> </mrow>
due to the fact thatIs symmetric, so there is an analytical solution:
<mrow> <msubsup> <mi>&amp;Theta;</mi> <mi>i</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <msup> <mi>&amp;eta;</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> <mi>Q</mi> <mrow> <mo>(</mo> <mi>D</mi> <mo>+</mo> <msqrt> <mrow> <msup> <mi>D</mi> <mn>2</mn> </msup> <mo>+</mo> <mn>4</mn> <msup> <mi>&amp;eta;</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mi>I</mi> </mrow> </msqrt> <mo>)</mo> </mrow> <msup> <mi>Q</mi> <mi>T</mi> </msup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow>
wherein, QDQTIs thatThe characteristics of (1).
5. Z-update (iv) according to claim 1, characterized in that Z-update can be divided into two parts: z0It refers to the "Θ |" that enforces sparsity in the inverse covariance matrixod,1-a penalty term; (Z)1,Z2) ψ -penalty term representing minimum cross-time deviation; these two updates can be resolved simultaneously, and each part can be parallelized to accelerate the computation.
6. Z according to claim 50Update, characterized in that each Zi,0Can be used asNear-end operator of norm with known closed-form solution
<mrow> <msubsup> <mi>Z</mi> <mrow> <mi>i</mi> <mo>,</mo> <mn>0</mn> </mrow> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>=</mo> <msub> <mi>prox</mi> <mrow> <mfrac> <mi>&amp;lambda;</mi> <mi>&amp;rho;</mi> </mfrac> <mo>|</mo> <mo>|</mo> <mo>&amp;CenterDot;</mo> <mo>|</mo> <msub> <mo>|</mo> <mrow> <mi>o</mi> <mi>d</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mrow> <mo>(</mo> <msubsup> <mi>&amp;Theta;</mi> <mi>i</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>U</mi> <mrow> <mi>i</mi> <mo>,</mo> <mn>0</mn> </mrow> <mi>k</mi> </msubsup> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>S</mi> <mfrac> <mi>&amp;lambda;</mi> <mi>&amp;rho;</mi> </mfrac> </msub> <mrow> <mo>(</mo> <msubsup> <mi>&amp;Theta;</mi> <mi>i</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>U</mi> <mrow> <mi>i</mi> <mo>,</mo> <mn>0</mn> </mrow> <mi>k</mi> </msubsup> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow>
Wherein,is an element intelligence-threshold function, i ≠ j ≠ p ≤ 1;
the (i, j) th element of this update is above.
7. (Z) according to claim 51,Z2) Update, characterized in that Zi-1,1And Zi,2To enhance lagrangian ties together, and therefore must be jointly updated; to obtain a closed form solution, define:
<mrow> <mover> <mi>&amp;psi;</mi> <mo>~</mo> </mover> <mo>(</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>Z</mi> <mn>1</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>Z</mi> <mn>2</mn> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>)</mo> <mo>=</mo> <mi>&amp;psi;</mi> <mrow> <mo>(</mo> <msub> <mi>Z</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>Z</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mrow>
for each (Z)1,Z2) In order to resolve a single update, the update is,
<mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msubsup> <mi>Z</mi> <mrow> <mi>i</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>Z</mi> <mrow> <mi>i</mi> <mo>,</mo> <mn>2</mn> </mrow> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <msub> <mi>prox</mi> <mrow> <mfrac> <mi>&amp;beta;</mi> <mi>&amp;rho;</mi> </mfrac> <mover> <mi>&amp;psi;</mi> <mo>~</mo> </mover> <mrow> <mo>(</mo> <mo>&amp;CenterDot;</mo> <mo>)</mo> </mrow> </mrow> </msub> <mo>(</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msubsup> <mi>&amp;Theta;</mi> <mrow> <mi>i</mi> <mo>-</mo> <mn>1</mn> </mrow> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>U</mi> <mrow> <mi>i</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> <mi>k</mi> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>&amp;Theta;</mi> <mi>i</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>U</mi> <mrow> <mi>i</mi> <mo>,</mo> <mn>2</mn> </mrow> <mi>k</mi> </msubsup> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>)</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow>
the results are shown in the above formula.
8. (Z) based on the column specification sum of claim 71,Z2) Update, characterized in that, orderg＝ψ，C＝[-I I]D is 0 andwill (Z)1,Z2) Update conversion to
<mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msubsup> <mi>Z</mi> <mrow> <mi>i</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>Z</mi> <mrow> <mi>i</mi> <mo>,</mo> <mn>2</mn> </mrow> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msubsup> <mi>&amp;Theta;</mi> <mrow> <mi>i</mi> <mo>-</mo> <mn>1</mn> </mrow> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>&amp;Theta;</mi> <mi>i</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>U</mi> <mrow> <mi>i</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> <mi>k</mi> </msubsup> <mo>+</mo> <msubsup> <mi>U</mi> <mrow> <mi>i</mi> <mo>,</mo> <mn>2</mn> </mrow> <mi>k</mi> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>&amp;Theta;</mi> <mrow> <mi>i</mi> <mo>-</mo> <mn>1</mn> </mrow> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>&amp;Theta;</mi> <mi>i</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>U</mi> <mrow> <mi>i</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> <mi>k</mi> </msubsup> <mo>+</mo> <msubsup> <mi>U</mi> <mrow> <mi>i</mi> <mo>,</mo> <mn>2</mn> </mrow> <mi>k</mi> </msubsup> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mo>-</mo> <mi>E</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>E</mi> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>9</mn> <mo>)</mo> </mrow> </mrow>
Wherein,due to the fact thatAndψ is the sum of column norms, so that E can be divided and simplified
[E]j＝(proxηψ(A))j＝proxηφ([A]j) (10)
Wherein phi isAndthe column norm of (d); will (Z) in the formula (4)i-1,1,Zi,2) The update is narrowed to find E in equation (11)jThe approximation operator of phi, defined by vectors, indicates that each vector has a closed form solution.
9. Closed form solution according to claim 8, characterized in that the intelligent product of elements is a product of the intelligence of the elementsIn the penalty term, the number of the first time,the near-end operator of isThe soft thresholds for the element intelligence product are as follows:
combined lassoIn a penalty term, used forThe near-end operator of the norm is a modular soft threshold,
for laplace penalty termNorm, the near-end operator of Laplace regularization is Re-editing into E in elemental formij＝(1+2η)-1(Aij)；
In the penalty term, the number of the first time,the near-end operator of the norm is
Wherein σ isThe solution of (1).
10. Perturbed node penalty term (Z) based on claim 71,Z2) Updating, characterized in that the variable (Z) is minimizedi-1,1,Zi,2) Is represented by (Y)1,Y2) (ii) a Then an additional variable V ═ W is introducedTIncreased LagrangianBecome into
Wherein,is a scaled bivariate, ρ is the same ADMM penalty parameter as the external ADMM; in the l-th iteration, three steps in the ADMM update are as follows:
(a)
(b)
(c)
whereinC＝[I -I I]，
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