          {  }@­We consider a linear inverse problem from the generative model:=observation++component 1 

  +component Kcomponent 2  We further assume:  Each component has the low-rank structure under reshuffling.         Reshuffling is an extension of tensor unfolding.
mode-3mode-2mode-2mode-3mode-2Incoherence measurement for each component:mode-2µi (A) := maxMode-1 Unfoldingmode-1mode-1j=ihigh rankReThe correspondenceglinuff

sh“Restore the Rubix Cube”Mode-3 Unfolding

Y ∈ TP (k ) (A),i iRi (Y)2 ≤ 1Rj (Y)2 ,(Theorem (Exact-Recovery condition): The minimizer of Reshuffled-TDequals the true components from the generative model when(b) The unfolded matrix X along the frist mode(a) a 3rd-order tensor with the size 3x3x3max

1.
max µi (Ai ) <i=1,...,N3K − 2Exact Recovery of Low-rank Tensor Decompositionunder Reshufflinglow rank(c) The matrix Y by the propsoed tensor reshufflingChao Li, Mohammad Emtiyaz Khan, Zhun Sun, Qibin Zhao∗1. We consider both cover and secret image as the components.
RIKEN Center for Advanced Intelligence ProjectNihonbashi, Chuo-ku, Tokyo 103-0027, Japan2. Our theory guarantees that the secret image can be reconstructed.
E-mail: {chao.li, emtiyaz.khan, zhun.sun, qibin.zhao}@riken.jpReshuffled-TD:covercontainersum upreshuffling operationAlgorithm 1 Reshuffled-TD via ADMMInitialize: The observation X ,the Lagrangian dual Y = sgn(X ),Ai = X /K, ∀i, and ρ > 1, κ0 > 0Iteration until convergence://STEP 1: update every component Ai , i = 1, . . . , Nfor i = 1, . . . , K do −1
Ai ← Ri−1 Dκ−1 Ri X − NA+κYjj=iend for//STEP 2: updatethe Lagrangiandual YKY ← Y + κ X − i=1 Ai//STEP 3: update the scalar κκ ← ρκOutput: (Ai , . . . , AN )(b) ObservationRReshuffled-TDsecretGreconstructionB(a) Generation (from components to observation)denotes matrix soft-thresholding.
Table 1: Performance comparison of image steganography.
In the experiment, we use SIR­ of SIRindicates better performance.

   (c) Decomposition (inverse problem)µi (A)(dB):= maxmaxRj (Y), cover(C) and secret(S) images, where larger valuetoquantifythedistortionofboth2j=i Y ∈ TPi (ki ) (A),(1)Ri (Y)2 ≤ 1Example 1LSBThe corresponding author

1Example 2∗cover/secretLSBDeep Stego Reshuffled-TDReshuffled-TDDatasets1 bit/ 2 bits/chnchn

σ=0.01

σ=0.05σ=0.1DTD(C)CART.(S)26.706.929.6614.4232.2513.1318.4522.6712.6325.92DTD(C)DTD(S)23.773.387.537.8436.864.7123.6911.3617.7813.12DTD(C)FIVEK(S)24.051.127.766.0034.304.9121.4711.6215.6413.54FIVEK(C)FIVEK(S)23.023.376.567.5236.33-0.9523.425.7017.499.29FVC(C)FIVEK(S)18.193.323.276.4233.545.0420.2512.8014.4014.66LIVE(C)FIVEK(S)24.504.087.667.5837.714.7524.7111.4918.7812.91Average23.373.707.078.3035.165 22.00 16.125.26 12.61 14.91Institute: Center for Artificial Intelligence ProjectGroup: Tensor Learning UnitRepresentor: Chao Li, Postdoc ResearcherInterests:machine learning, signal processing, computer vision1Contact: chao.li@riken.jp