| Notations | Descriptions | Notations | Descriptions |
| a, b, c, ... | calvars | a, b, c, ... | vectors |
| A, B, C, ... | matrices | A, B, C, ... | tensors |
| A, B, C, ... | sets | A, B, C, ... | operators |
| [1:n] | {1,2,...,n} | AT | transpose of A |
| I | identity tensor | 0 | null tensor |
| [At1,i2,...,ih] | (i1,i2,...,ih)-th element of A | ||A||0 | the number of non-zero elements of A |
| ||A||1 | ||A||1 = ∑i1,i2,...,ih ||A||i1,i2,...,ih| | ATk1,...,kh | rotation of A ∈ R1×...×Ih such that ATk1,...,kh ∈ Rl2×...×Ikh |
| [Atk1,...,kh], ...,ik3,...,ikh | slice along the (k1,k2)-th mode | A(k) (or [Aj(n)]) | Mode-n Unfolding of A |
| ×n | Mode-n product | Ac | complementary set of A |
",
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+ "type": "text",
+ "text": "2.1 Tensor Completion Problem",
+ "text_level": 1,
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+ "type": "text",
+ "text": "Considering tensor data $\\mathcal{M} \\in \\mathbb{R}^{I_1 \\times I_2 \\times \\dots \\times I_h}$ may with missing elements, [17, 20] propose the following t-SVD-based tensor completion model:",
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+ "type": "equation",
+ "text": "\n$$\n\\min _ {\\boldsymbol {x}} \\operatorname {r a n k} _ {[ k _ {1}, k _ {2} ]} (\\boldsymbol {x}) \\quad s. t. \\Psi_ {\\mathbb {I}} (\\boldsymbol {\\mathcal {M}}) = \\Psi_ {\\mathbb {I}} (\\boldsymbol {x}), \\tag {1}\n$$\n",
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+ "type": "text",
+ "text": "where $\\varPsi_{\\mathbb{I}}$ is a linear project operator on the support set $\\mathbb{I}$ such that",
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+ "type": "equation",
+ "text": "\n$$\n[ \\varPsi_{\\mathbb{I}}(\\mathcal{M})]_{i_{1},i_{2},\\dots ,i_{h}}=\\left\\{\\begin{array}{ll}[\\mathcal{M}]_{i_{1},i_{2},\\cdots,i_{h}},& \\mbox{if } (i_{1},i_{2},\\cdots,i_{h})\\in\\mathbb{I};\\\\ 0,&\\mbox{if }(i_{1},i_{2},\\cdots,i_{h})\\notin\\mathbb{I},\\end{array}\\right.\n$$\n",
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+ {
+ "type": "text",
+ "text": "and $i_k \\in [1: I_k]$ for $k = 1, 2, \\dots, h$ . The tensor rank function in (1) operates on the assumption that an $h$ -order tensor data $\\mathcal{M} \\in \\mathbb{R}^{I_1 \\times I_2 \\times \\dots \\times I_h}$ in the real world can be decomposed as",
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+ "type": "equation",
+ "text": "\n$$\n\\boldsymbol {\\mathcal {M}} = \\boldsymbol {\\mathcal {Z}} \\times_ {k _ {3}} \\hat {\\boldsymbol {U}} _ {k _ {3}} ^ {T} \\dots \\times_ {k _ {h}} \\hat {\\boldsymbol {U}} _ {k _ {h}} ^ {T}, \\tag {2}\n$$\n",
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+ "type": "text",
+ "text": "where $\\{k_3, k_4, \\dots, k_h\\} \\subset [1:h]$ , $\\{\\hat{U}_{k_n}\\}_{n=3}^h$ are a set of given invertible transforms, and",
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+ "type": "equation",
+ "text": "\n$$\n\\begin{array}{l} \\operatorname {r a n k} _ {[ k _ {1}, k _ {2} ]} (\\boldsymbol {\\mathcal {M}}) = \\max _ {i _ {k _ {n}} \\in [ 1: I _ {k _ {n}} ] \\text {f o r} n = 3, 4, \\dots , h} \\operatorname {r a n k} \\left(\\left[ \\boldsymbol {\\mathcal {Z}} ^ {T _ {k _ {1}}, \\dots , k _ {h}} \\right] _ {:,: i _ {k _ {3}}, \\dots , i _ {k _ {h}}}\\right) \\\\ < < \\min \\left(I _ {k _ {1}}, I _ {k _ {2}}\\right) \\tag {3} \\\\ \\end{array}\n$$\n",
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+ "type": "text",
+ "text": "for certain $(k_{1},k_{2})$ satisfying $1\\leq k_{1} < k_{2}\\leq h$ . If taken $\\{\\hat{U}_{k_n}\\}_{n = 3}^h$ as Discrete Fourier Matrices, $\\mathrm{rank}_{[k_1,k_2]}(\\mathcal{M})$ is referred as to tensor tubal rank of $\\mathcal{M}$ [16].",
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+ "type": "text",
+ "text": "2.2 The Proposed Tensor Completion with A Learnable Tensor Rank Function",
+ "text_level": 1,
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+ "type": "text",
+ "text": "From the definition of $\\mathrm{rank}_{[k_1,k_2]}(\\mathcal{M})$ , we can see that it aims to examine the slice-wise tensor rank along the $(k_{1},k_{2})$ -th mode of transformed data $\\pmb{\\mathcal{Z}}$ . It allows (1) to investigate the low-rankness of different features in tensor data corresponding to different frequencies separately and jointly.",
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+ "type": "page_number",
+ "text": "4",
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+ "type": "header",
+ "text": "J. Zheng et al.",
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+ "type": "text",
+ "text": "However, when $\\mathcal{M}$ exhibits harsh changes caused by disordered tensor slices along a certain mode $k_{n}$ , all information in $\\mathcal{M}$ tends to collapse into the high-frequency slices in $\\mathcal{Z}$ . To handle such situation, we introduce a learnable permutation matrix $P\\in \\mathbb{R}^{I_{k_n}\\times I_{k_n}}$ to (2):",
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+ "type": "equation",
+ "text": "\n$$\n\\pmb {\\mathcal {M}} \\times_ {k _ {n}} \\pmb {P} = \\pmb {\\mathcal {Z}} \\times_ {k _ {3}} \\hat {\\pmb {U}} _ {k _ {3}} ^ {T} \\dots \\times_ {k _ {n}} (\\hat {\\pmb {U}} _ {k _ {n}} \\pmb {P}) ^ {T} \\times_ {k _ {n + 1}} \\dots \\times_ {k _ {h}} \\hat {\\pmb {U}} _ {k _ {h}} ^ {T},\n$$\n",
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+ {
+ "type": "text",
+ "text": "and therefore we can get the following model:",
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+ "type": "equation",
+ "text": "\n$$\n\\begin{array}{l} \\min _ {\\boldsymbol {X}, \\boldsymbol {P} _ {k _ {n}}} \\operatorname {r a n k} _ {[ k _ {1}, k _ {2} ]} (\\boldsymbol {X} \\times_ {k _ {n}} \\boldsymbol {P} _ {k _ {n}}) \\\\ s. t. \\Psi_ {\\mathbb {I}} (\\boldsymbol {\\mathcal {M}}) = \\Psi_ {\\mathbb {I}} (\\boldsymbol {\\mathcal {X}}), \\\\ \\sum_ {i = 1} ^ {I _ {k _ {n}}} \\left[ \\boldsymbol {P} _ {k _ {n}} \\right] _ {i, j} = 1, \\quad \\sum_ {j = 1} ^ {I _ {k _ {n}}} \\left[ \\boldsymbol {P} _ {k _ {n}} \\right] _ {i, j} = 1 \\text {f o r} \\left[ \\boldsymbol {P} _ {k _ {n}} \\right] _ {i, j} \\in \\{0, 1 \\}. \\tag {4} \\\\ \\end{array}\n$$\n",
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+ "type": "text",
+ "text": "This incorporation is aimed at addressing the slice permutation property in tensor completion methods, thus facilitating a more effective exploration of the low-rank property in $\\mathcal{M}$ .",
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+ "type": "text",
+ "text": "Unfortunately, solving (4) is challenging due to the constraints of $\\sum_{i=1}^{I_{k_n}}[P_{k_n}]_{i,j} = 1$ and $\\sum_{j=1}^{I_{k_n}}[P_{k_n}]_{i,j} = 1$ for $[P_{k_n}]_{i,j} \\in \\{0,1\\}$ . Therefore, we opt to use a set of learnable unitary matrices $\\{\\pmb{U}_{k_n}\\}_{n=s+1}^h$ instead, proposing the following tensor completion model for given $\\{k_{s+1}, k_{s+2}, \\dots, k_h\\}$ :",
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+ "type": "equation",
+ "text": "\n$$\n\\begin{array}{l} \\min _ {\\boldsymbol {X}, \\boldsymbol {U} _ {k _ {n}} (n = s + 1, \\dots , h)} \\operatorname {r a n k} _ {[ k _ {1}, k _ {2} ]} (\\boldsymbol {\\mathcal {X}} \\times_ {k _ {s + 1}} \\boldsymbol {U} _ {k _ {s + 1}} \\dots \\times_ {k _ {h}} \\boldsymbol {U} _ {k _ {h}}) \\\\ s. t. \\Psi_ {\\mathbb {I}} (\\boldsymbol {\\mathcal {M}}) = \\Psi_ {\\mathbb {I}} (\\boldsymbol {\\mathcal {X}}), \\boldsymbol {U} _ {k _ {n}} ^ {T} \\boldsymbol {U} _ {k _ {n}} = \\boldsymbol {I} (n = s + 1, \\dots , h), \\tag {5} \\\\ \\end{array}\n$$\n",
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+ "type": "text",
+ "text": "where $\\mathcal{X}$ is a low-rank estimation of the true tensor data $\\mathcal{M}$ . Besides simplifying the optimization process of (4), the introduction of the learnable unitary matrices enables the model to extract features for better studying the low-rankness in tensor data and to handle other scenarios where the initially provided transforms $\\{\\hat{U}_{k_n}\\}_{n=3}^h$ are inadequate, such as videos with irregularly changing content and image sequences with different scenes. Hence, we introduce our learnable tensor rank as",
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+ "type": "equation",
+ "text": "\n$$\n\\mathrm {r a n k} _ {[ k _ {1}, k _ {2} ], \\tilde {\\mathcal {U}}} (\\pmb {\\mathscr {X}}) = \\mathrm {r a n k} _ {[ k _ {1}, k _ {2} ]} (\\tilde {\\mathcal {U}} (\\pmb {\\mathscr {X}})),\n$$\n",
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+ "type": "text",
+ "text": "where $\\tilde{\\mathcal{U}} (\\pmb {x}) = \\pmb {x}\\times_{k_{s + 1}}\\pmb{U}_{k_{s + 1}}\\dots \\times_{k_h}\\pmb{U}_{k_h}.$",
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+ "type": "text",
+ "text": "2.3 Approximation to The Proposed Tensor Completion by Using A Learnable Tensor Nuclear Norm (TC-SL)",
+ "text_level": 1,
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+ "type": "text",
+ "text": "Since the function $\\mathrm{rank}_{[k_1,k_2],\\hat{\\mathcal{U}}}(\\cdot)$ is discrete, it often leads to the NP-hard problem. From the result given in [16,17], we know that, when $h = 3$ and $\\hat{U}_3$ is orthogonal, $\\| \\cdot \\|_{*}^{[1,2]}$ is the tightest convex envelope of $\\mathrm{rank}_{[1,2]}(\\cdot)$ on the set $\\{\\mathcal{A}||\\mathcal{A}||_2^{[1,2]}\\leq 1\\}$ , where the definitions of $\\| \\cdot \\| _2^{[k_1,k_2]}$ and $\\| \\cdot \\|_{*}^{[k_{1},k_{2}]}$ are given in Definition 1. This conclusion can be easily extended to our case, and we obtain the Property 1.",
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+ {
+ "type": "header",
+ "text": "Handling The Non-Smooth Challenge in Tensor SVD",
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+ "type": "page_number",
+ "text": "5",
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+ "type": "text",
+ "text": "Definition 1. For an $h$ -order tensor $\\mathcal{A} \\in \\mathbb{R}^{I_1 \\times I_2 \\times \\dots \\times I_h}$ , $\\| \\mathcal{A} \\|_*^{[k_1, k_2]}$ and $\\| \\mathcal{A} \\|_2^{[k_1, k_2]}$ are defined as",
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+ "type": "equation",
+ "text": "\n$$\n\\| \\boldsymbol {\\mathcal {A}} \\| _ {*} ^ {[ k _ {1}, k _ {2} ]} = \\sum_ {i _ {k _ {3}}, i _ {k _ {4}}, \\dots , i _ {k _ {h}}} \\| [ \\boldsymbol {\\mathcal {A}} ^ {T _ {k _ {1}}, \\dots , k _ {h}} \\times_ {k _ {3}} \\hat {\\boldsymbol {U}} _ {k _ {3}} \\cdot \\cdot \\cdot \\times_ {k _ {h}} \\hat {\\boldsymbol {U}} _ {k _ {h}} ] _ {:;,: i _ {k _ {3}}, \\dots , i _ {k _ {h}}} \\| _ {*}\n$$\n",
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+ {
+ "type": "text",
+ "text": "and",
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+ "type": "equation",
+ "text": "\n$$\n\\| \\pmb {\\mathcal {A}} \\| _ {2} ^ {[ k _ {1}, k _ {2} ]} = \\max _ {i _ {k _ {3}}, i _ {k _ {4}}, \\dots , i _ {k _ {h}}} \\| [ \\pmb {\\mathcal {A}} ^ {T _ {k _ {1}}, \\dots , k _ {h}} \\times_ {k _ {3}} \\hat {\\pmb {U}} _ {k _ {3}} \\cdot \\cdot \\cdot \\times_ {k _ {h}} \\hat {\\pmb {U}} _ {k _ {h}} ] _ {:,: i _ {k _ {3}}, \\dots , i _ {k _ {h}}} \\| _ {2},\n$$\n",
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+ "type": "text",
+ "text": "respectively.",
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+ "type": "text",
+ "text": "Property 1. For an h-order tensor $\\mathcal{A} \\in \\mathbb{R}^{I_1 \\times I_2 \\times \\dots \\times I_h}$ , let us define $\\| \\mathcal{A} \\|_{*,\\tilde{\\mathcal{U}}}^{[k_1,k_2]}$ and $\\| \\mathcal{A} \\|_{2,\\tilde{\\mathcal{U}}}^{[k_1,k_2]}$ are defined as $\\| \\mathcal{A} \\|_{*,\\tilde{\\mathcal{U}}}^{[k_1,k_2]} = \\| \\tilde{\\mathcal{U}} (\\mathcal{A}) \\|_{*}^{[k_1,k_2]}$ and $\\| \\mathcal{A} \\|_{2,\\tilde{\\mathcal{U}}}^{[k_1,k_2]} = \\| \\tilde{\\mathcal{U}} (\\mathcal{A}) \\|_{2}^{[k_1,k_2]}$ , respectively. $\\| \\cdot \\|_{*,\\tilde{\\mathcal{U}}}^{[k_1,k_2]}$ is the dual norm of tensor $\\| \\cdot \\|_{2,\\tilde{\\mathcal{U}}}^{[k_1,k_2]}$ norm, and $\\| \\cdot \\|_{*,\\tilde{\\mathcal{U}}}^{[k_1,k_2]}$ is the tightest convex envelope of $\\mathrm{rank}_{[k_1,k_2],\\tilde{\\mathcal{U}}}(\\cdot)$ on the set $\\{\\mathcal{A} \\| \\mathcal{A} \\|_{2,\\tilde{\\mathcal{U}}}^{[k_1,k_2]} \\leq 1\\}$ .",
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+ "type": "text",
+ "text": "Therefore, we derive an approximation to the proposed model (6) by utilizing a learnable tensor nuclear norm based on the prior assumption of slice-wise low-rankness in the transformed data (TC-SL):",
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+ "type": "equation",
+ "text": "\n$$\n\\min _ {\\boldsymbol {X}, \\tilde {\\mathcal {U}}} \\| \\boldsymbol {X} \\| _ {*}, \\tilde {\\mathcal {U}} ^ {[ k _ {1}, k _ {2} ]} \\quad s. t. \\Psi_ {\\mathbb {I}} (\\boldsymbol {\\mathcal {M}}) = \\Psi_ {\\mathbb {I}} (\\boldsymbol {\\mathcal {X}}). \\tag {6}\n$$\n",
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+ "type": "text",
+ "text": "Let $\\hat{\\mathcal{U}}$ be the invertible transform operator learned by (6). The exactly recovery of TC-SL with given $\\hat{\\mathcal{U}}$ is guaranteed from the current studies in the exactly recovery of the t-SVD-based tensor completion [16, 17].",
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+ "type": "text",
+ "text": "2.4 The Proposed Multi-Objective Tensor Completion to Learn The Cross-Dimensional Low-Rankness",
+ "text_level": 1,
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+ "type": "text",
+ "text": "From (6), we observe that the definition of TC-SL depends on the choice of $k_{1}$ and $k_{2}$ , and it considers different kinds of low-rankness in the tensor data by adjusting $(k_{1}, k_{2})$ . However, considering only one mode may result in the loss of correlation information across the remaining modes. To address this issue, we give the following multi-objective model with learnable tensor nuclear norms for tensor completion (MOTC):",
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+ "type": "equation",
+ "text": "\n$$\n\\min _ {\\boldsymbol {x}, \\tilde {\\mathcal {U}} _ {\\left(k _ {1}, k _ {2}\\right)}} \\left[ \\| \\boldsymbol {\\mathcal {X}} \\| _ {*}, \\tilde {\\mathcal {U}} _ {\\left(k _ {1}, k _ {2}\\right)} ^ {[ k _ {1}, k _ {2} ]} \\right] _ {1 \\leq k _ {1} < k _ {2} \\leq h} \\quad s. t. \\Psi_ {\\mathbb {I}} (\\boldsymbol {\\mathcal {M}}) = \\Psi_ {\\mathbb {I}} (\\boldsymbol {\\mathcal {X}}), \\tag {7}\n$$\n",
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+ "type": "text",
+ "text": "where $\\tilde{\\mathcal{U}}_{(k_1,k_2)}(\\boldsymbol {x}) = \\boldsymbol {x}\\times_{k_{s + 1}}\\boldsymbol{U}_{k_{s + 1}}^{(k_1,k_2)}\\dots \\times_{k_h}\\boldsymbol{U}_{k_h}^{(k_1,k_2)}$ and $(\\boldsymbol{U}_{k_n}^{(k_1,k_2)})^T\\boldsymbol{U}_{k_n}^{(k_1,k_2)} = \\boldsymbol{I}$ for $n = 1 + s,\\dots ,h$ and $1\\leq k_{1} < k_{2}\\leq h$ . In MOTC, the multiple objective functions $\\| \\pmb {x}\\|_{*,\\tilde{\\mathcal{U}}_{(k_1,k_2)}}^{[k_1,k_2]}$ $(1\\leq k_{1} < k_{2}\\leq h)$ are utilized to examine the low-rankness of tensor data from its various dimensions.",
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+ "type": "page_number",
+ "text": "6",
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+ "type": "header",
+ "text": "J. Zheng et al.",
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+ },
+ {
+ "type": "text",
+ "text": "3 Optimization Algorithm",
+ "text_level": 1,
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+ "type": "text",
+ "text": "In this section, we provide a detailed discussion for the optimizing (6) and (7).",
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+ "type": "text",
+ "text": "3.1 Alternating Proximal Multiplier Method (APMM) for TC-SL",
+ "text_level": 1,
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+ "type": "text",
+ "text": "To solve the problem (6), we introduce auxiliary variables $\\pmb{\\mathcal{E}}\\in \\mathbb{E} = \\{\\pmb {\\mathcal{E}}|\\Psi_{\\mathbb{I}}(\\pmb {\\mathcal{E}}) = \\mathbf{0}\\}$ and $\\pmb{\\Xi}$ such that $\\pmb {x} = \\pmb {z}\\times_{k_{s + 1}}\\pmb{U}_{k_{s + 1}}^T\\dots \\times_{k_h}\\pmb{U}_{k_h}^T$ . Therefore, we turn to solve the following equivalence problem:",
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+ "type": "equation",
+ "text": "\n$$\n\\min _ {\\boldsymbol {Z}, \\boldsymbol {U} _ {k _ {n}} ^ {T} \\boldsymbol {U} _ {k _ {n}} = \\boldsymbol {I} (n = s + 1, \\dots , h)} \\| \\boldsymbol {Z} \\| _ {*} ^ {[ k _ {1}, k _ {2} ]} \\quad \\text {s . t .} \\Psi_ {\\mathbb {I}} (\\boldsymbol {\\mathcal {M}}) = \\boldsymbol {\\mathcal {Z}} \\times_ {k _ {s + 1}} \\boldsymbol {U} _ {k _ {s + 1}} ^ {T} \\dots \\times_ {k _ {h}} \\boldsymbol {U} _ {k _ {h}} ^ {T} + \\boldsymbol {\\mathcal {E}}. \\tag {8}\n$$\n",
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+ "type": "text",
+ "text": "The Augmented Lagrangian function of (8) is formulated as",
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+ "type": "equation",
+ "text": "\n$$\n\\begin{array}{l} \\mathcal {L} (\\boldsymbol {\\mathcal {Z}}, \\{\\boldsymbol {U} _ {k _ {n}} \\} _ {n = s + 1} ^ {h}, \\boldsymbol {\\mathcal {E}}, \\boldsymbol {\\mathcal {Y}}, \\mu) = \\| \\boldsymbol {\\mathcal {Z}} \\| _ {*} ^ {[ k _ {1}, k _ {2} ]} + \\left\\langle \\Psi_ {\\mathbb {I}} (\\boldsymbol {\\mathcal {M}}) - \\boldsymbol {\\mathcal {Z}} \\times_ {k _ {s + 1}} \\boldsymbol {U} _ {k _ {s + 1}} ^ {T} \\dots \\times_ {k _ {h}} \\boldsymbol {U} _ {k _ {h}} ^ {T} \\right. \\\\ - \\mathcal {E}, \\mathcal {Y} \\rangle + \\frac {\\mu}{2} \\| \\Psi_ {\\mathbb {I}} (\\mathcal {M}) - \\mathcal {Z} \\times_ {k _ {s + 1}} \\boldsymbol {U} _ {k _ {s + 1}} ^ {T} \\dots \\times_ {k _ {h}} \\boldsymbol {U} _ {k _ {h}} ^ {T} - \\mathcal {E} \\| _ {F} ^ {2}, \\tag {9} \\\\ \\end{array}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 6
+ },
+ {
+ "type": "text",
+ "text": "where $\\mathcal{V}$ is Lagrange multiplier, and $\\mu$ is a positive scalar. We solve (8) iteratively by combining the proximal algorithm with the Alternating Direction Method of Multipliers (APMM) that is given in the Algorithm 1. We detail the solutions for solving $\\mathcal{Z}^{(t + 1)}, U_{k_n}^{(t + 1)}$ , and $\\mathcal{E}^{(t + 1)}$ as follows.",
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "Calculate $\\mathcal{Z}^{(t + 1)}$",
+ "bbox": [
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+ "page_idx": 6
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\begin{array}{l} \\boldsymbol {\\mathcal {Z}} ^ {(t + 1)} = \\underset {\\boldsymbol {\\mathcal {Z}}} {\\arg \\min } \\frac {1}{2} \\| \\frac {\\mu^ {(t)} \\hat {\\boldsymbol {\\mathcal {P}}} \\times_ {k _ {s + 1}} \\boldsymbol {U} _ {k _ {s + 1}} ^ {(t)} \\cdots \\times_ {k _ {h}} \\boldsymbol {U} _ {k _ {h}} ^ {(t)} + \\eta^ {(t)} \\boldsymbol {\\mathcal {Z}} ^ {(t)}}{\\mu^ {(t)} + \\eta^ {(t)}} - \\boldsymbol {\\mathcal {Z}} \\| _ {F} ^ {2} \\\\ + \\frac {1}{\\mu^ {(t)} + \\eta^ {(t)}} \\| \\mathcal {Z} \\| _ {*} ^ {[ k _ {1}, k _ {2} ]}, \\\\ \\end{array}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 6
+ },
+ {
+ "type": "text",
+ "text": "where $\\hat{\\mathcal{P}} = \\Psi_{\\mathbb{I}}(\\mathcal{M}) - \\pmb{\\mathcal{E}}^{(t)} + \\frac{1}{\\mu^{(t)}}\\pmb{\\mathcal{Y}}^{(t)}$ . It can be solved by the tensor singular value thresholding operation with parameter $\\frac{1}{\\mu^{(t)} + \\eta^{(t)}}$ [15].",
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "Calculate $U_{k_n}^{(t + 1)}$ : Let $\\mathcal{A} = \\mathcal{Z}^{(t + 1)} \\times_{k_h} U_{k_h}^{(t)T} \\times_{k_h - 1} \\dots \\times_{k_n + 1} U_{k_n + 1}^{(t)T}$ and $\\mathcal{B} = \\hat{\\mathcal{P}} \\times_{s + 1} U_{k_{s + 1}}^{(t + 1)} \\dots \\times_{k_n - 1} U_{k_n - 1}^{(t + 1)}$ , and we have",
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+ {
+ "type": "equation",
+ "text": "\n$$\n\\boldsymbol {U} _ {k _ {n}} ^ {(t + 1)} = \\underset {\\boldsymbol {U} _ {k _ {n}} ^ {T} \\boldsymbol {U} _ {k _ {n}} = \\boldsymbol {I}} {\\arg \\min } \\| \\boldsymbol {U} _ {k _ {n}} [ \\sqrt {\\mu^ {(t)}} \\boldsymbol {\\mathcal {B}} _ {(k _ {n})}, \\sqrt {\\eta^ {(t)}} \\boldsymbol {I} ] - [ \\sqrt {\\mu^ {(t)}} \\boldsymbol {\\mathcal {A}} _ {(k _ {n})}, \\sqrt {\\eta^ {(t)}} \\boldsymbol {U} _ {k _ {n}} ^ {(t)} ] \\| _ {F} ^ {2}.\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ {
+ "type": "text",
+ "text": "The optimal solution $\\pmb{U}_{k_n}^{(t + 1)}$ can be gievn by $\\pmb{U}_{k_n}^{(t + 1)} = \\pmb{U}\\pmb{V}^T$ from [29], where $\\pmb{U}$ and $\\pmb{V}$ can be obtained by SVD of $\\mu^{(t)}\\pmb{A}_{(k_n)}\\pmb{B}_{(k_n)}^T + \\eta^{(t)}\\pmb{U}_{k_n}^{(t)}$ :",
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+ {
+ "type": "equation",
+ "text": "\n$$\n\\boldsymbol {\\mu} ^ {(t)} \\boldsymbol {\\mathcal {A}} _ {(k _ {n})} \\boldsymbol {\\mathcal {B}} _ {(k _ {n})} ^ {T} + \\eta^ {(t)} \\boldsymbol {U} _ {k _ {n}} ^ {(t)} = \\boldsymbol {U} \\boldsymbol {\\Sigma} \\boldsymbol {V} ^ {T}.\n$$\n",
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+ {
+ "type": "header",
+ "text": "Handling The Non-Smooth Challenge in Tensor SVD",
+ "bbox": [
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+ "type": "page_number",
+ "text": "7",
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+ {
+ "type": "text",
+ "text": "Algorithm 1: APMM-based Iterative Solver to (8)",
+ "bbox": [
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+ "page_idx": 7
+ },
+ {
+ "type": "text",
+ "text": "Input: $\\Psi_{\\mathbb{I}}(\\mathcal{M}),\\{U_{k_n}^{(0)}\\}_{n = s + 1}^h,\\{\\hat{U}_{k_n}^{(0)}\\}_{n = 3}^h,\\pmb {\\varepsilon}^{(0)},\\pmb {\\mathcal{Y}}^{(0)},t = 0,\\mu^{(0)},\\eta^{(0)}$",
+ "bbox": [
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+ ],
+ "page_idx": 7
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\rho_ {\\mu}, \\rho_ {\\eta} > 1, \\bar {\\mu}, \\text {a n d} \\bar {\\eta}.\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 7
+ },
+ {
+ "type": "text",
+ "text": "Output: $\\mathcal{Z}^{(t + 1)},\\{U_{k_n}^{(t + 1)}\\}_{n = s + 1}^h$ , and $\\pmb{x}^{(t + 1)}$",
+ "bbox": [
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+ "page_idx": 7
+ },
+ {
+ "type": "list",
+ "sub_type": "text",
+ "list_items": [
+ "1. While not converge do",
+ "2. $\\pmb{\\mathcal{Z}}^{(t + 1)} = \\arg \\min_{\\pmb{\\mathcal{Z}}}\\mathcal{L}(\\pmb {\\mathcal{Z}},\\{\\pmb{U}_{k_n}^{(t)}\\}_{n = s + 1}^h,\\pmb{\\mathcal{E}}^{(t)},\\pmb{\\mathcal{Y}}^{(t)},\\mu^{(t)}) + \\frac{\\eta^{(t)}}{2}\\| \\pmb{\\mathcal{Z}}^{(t)} - \\pmb {\\mathcal{Z}}\\| _F^2;$",
+ "3. Calculate $\\pmb{U}_{k_n}^{(t + 1)} = \\arg \\min_{\\pmb{U}_{k_n}^T\\pmb{U}_{k_n} = \\pmb{I}}\\mathcal{L}(\\pmb{\\mathcal{Z}}^{(t + 1)},\\{\\pmb{U}_{k_n}^{(t + 1)}\\}_{n = s + 1}^{n - 1},$"
+ ],
+ "bbox": [
+ 240,
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+ 754,
+ 277
+ ],
+ "page_idx": 7
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\boldsymbol {U} _ {k _ {n}}, \\{\\boldsymbol {U} _ {k _ {n}} ^ {(t)} \\} _ {n = n + 1} ^ {h}, \\boldsymbol {\\mathcal {E}} ^ {(t)}, \\boldsymbol {\\mathcal {Y}} ^ {(t)}, \\mu^ {(t)}) + \\frac {\\eta^ {(t)}}{2} \\| \\boldsymbol {U} _ {k _ {n}} ^ {(t)} - \\boldsymbol {U} _ {k _ {n}} \\| _ {F} ^ {2}, (s + 1 \\leq n _ {0} \\leq h);\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 240,
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+ 740,
+ 303
+ ],
+ "page_idx": 7
+ },
+ {
+ "type": "text",
+ "text": "4. Calculate $\\pmb{\\chi}^{(t + 1)}$ by $\\pmb{\\chi}^{(t + 1)} = \\pmb{Z}^{(t + 1)}\\times_{k_h}\\pmb{U}_{k_h}^{(t + 1)T}\\times_{k_{h - 1}}\\dots \\times_{k_{s + 1}}\\pmb{U}_{k_{s + 1}}^{(t + 1)T}$ ;",
+ "bbox": [
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+ "page_idx": 7
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\boldsymbol {\\mathcal {E}} ^ {(t + 1)} = \\arg \\min _ {\\boldsymbol {\\mathcal {E}} \\in \\mathbb {E}} \\mathcal {L} (\\boldsymbol {\\mathcal {Z}} ^ {(t + 1)}, \\{\\boldsymbol {U} _ {k _ {n}} ^ {(t + 1)} \\} _ {n = s + 1} ^ {h}, \\boldsymbol {\\mathcal {E}}, \\boldsymbol {\\mathcal {Y}} ^ {(t)}, \\mu^ {(t)}) + \\frac {\\eta^ {(t)}}{2} \\| \\boldsymbol {\\mathcal {E}} - \\boldsymbol {\\mathcal {E}} ^ {(t)} \\| _ {F} ^ {2};\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 7
+ },
+ {
+ "type": "list",
+ "sub_type": "text",
+ "list_items": [
+ "6. Calculate $\\pmb{y}^{(t + 1)}$ by $\\pmb{y}^{(t + 1)} = \\mu^{(t)}(\\varPsi_{\\mathbb{I}}(\\mathcal{M}) - \\pmb{x}^{(t + 1)} - \\pmb{\\varepsilon}^{(t + 1)}) + \\pmb{y}^{(t)}$",
+ "7. Calculate $\\mu^{(t + 1)} = \\min (\\bar{\\mu},\\rho_{\\mu}\\mu^{(t)})$ and $\\eta^{(t + 1)} = \\min (\\bar{\\eta},\\rho_{\\eta}\\eta^{(t)})$ , respectively;",
+ "8. Check the convergence condition: $\\| \\pmb{Z}^{(t + 1)} - \\pmb{Z}^{(t)}\\|_{\\infty} < \\varepsilon$"
+ ],
+ "bbox": [
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+ 398
+ ],
+ "page_idx": 7
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\| \\boldsymbol {\\mathcal {X}} ^ {(t + 1)} - \\boldsymbol {\\mathcal {X}} ^ {(t)} \\| _ {\\infty} < \\varepsilon ,\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 250,
+ 398,
+ 403,
+ 412
+ ],
+ "page_idx": 7
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\left\\| \\boldsymbol {U} _ {k _ {n}} ^ {(t + 1)} - \\boldsymbol {U} _ {k _ {n}} ^ {(t)} \\right\\| _ {\\infty} < \\varepsilon \\text {f o r} n = s + 1, s + 2, \\dots , h;\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 7
+ },
+ {
+ "type": "text",
+ "text": "9. $t = t + 1$",
+ "bbox": [
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+ "page_idx": 7
+ },
+ {
+ "type": "text",
+ "text": "10. end while",
+ "bbox": [
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+ ],
+ "page_idx": 7
+ },
+ {
+ "type": "text",
+ "text": "Calculate $\\mathcal{E}^{(t + 1)}$",
+ "bbox": [
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+ ],
+ "page_idx": 7
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\begin{array}{l} \\boldsymbol {\\mathcal {E}} ^ {(t + 1)} = \\underset {\\boldsymbol {\\mathcal {E}} \\in \\mathbb {E}} {\\arg \\min } \\frac {\\mu^ {(t)}}{2} \\| \\Psi_ {\\mathbb {I}} (\\boldsymbol {\\mathcal {M}}) - \\boldsymbol {\\mathcal {X}} ^ {(t + 1)} - \\boldsymbol {\\mathcal {E}} + \\frac {1}{\\mu^ {(t)}} \\boldsymbol {\\mathcal {Y}} ^ {(t)} \\| _ {F} ^ {2} + \\frac {\\eta^ {(t)}}{2} \\| \\boldsymbol {\\mathcal {E}} - \\boldsymbol {\\mathcal {E}} ^ {(t)} \\| _ {F} ^ {2} \\\\ = \\Psi_ {\\mathbb {I} ^ {c}} \\big (\\frac {1}{\\mu^ {(t)} + \\eta^ {(t)}} \\big (\\mu^ {(t)} (\\Psi_ {\\mathbb {I}} (\\boldsymbol {\\mathcal {M}}) - \\boldsymbol {\\mathcal {X}} ^ {(t + 1)} + \\frac {1}{\\mu^ {(t)}} \\boldsymbol {\\mathcal {Y}} ^ {(t)}) + \\eta^ {(t)} \\boldsymbol {\\mathcal {E}} ^ {(t)}) \\big). \\\\ \\end{array}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 7
+ },
+ {
+ "type": "text",
+ "text": "Computational Complexity The most time-consuming steps in the algorithm 1 are the computations of $\\mathcal{Z}$ , and $U_{k_n}$ . Since the computational complexity of n-mode product of $\\hat{\\mathcal{P}}\\in \\mathbb{R}^{I_1\\times I_2\\times \\dots \\times I_h}$ and $U_{n}\\in \\mathbb{R}^{I_{n}\\times I_{n}}$ is $\\mathcal{O}(I_nI_1I_2\\dots I_h)$ , the complexity of the computation of $\\mathcal{Z}$ is $\\mathcal{O}(hI_{(1)}I_1I_2\\dots I_h)$ , where $I_{(1)} = \\max_k(I_k)$ . Besides, the complexity of the computation of $U_{k_n}$ is $\\mathcal{O}((h - 1)I_{(1)}I_1I_2\\dots I_h + I_k^2 I_1I_2\\dots I_h)$ , therefore the overall computational complexity of each iteration of Algorithm 1 is $\\mathcal{O}\\Big((h - s)(h + I_{(1)} - 1)I_{(1)}I_1I_2\\dots I_h + hI_{(1)}I_1I_2\\dots I_h\\Big)$ .",
+ "bbox": [
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+ "page_idx": 7
+ },
+ {
+ "type": "text",
+ "text": "Convergence Analysis Although the optimization problem (8) is non-convex because of the constraints $\\pmb{U}_{k_n}^T\\pmb{U}_{k_n} = \\pmb{I}(n = s + 1,s + 2,\\dots ,h)$ and the global optimality for (8) is hardly guaranteed, we can still prove some excellent convergence properties of the Algorithm 1, as stated in the following theorem.",
+ "bbox": [
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+ "page_idx": 7
+ },
+ {
+ "type": "page_number",
+ "text": "8",
+ "bbox": [
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+ ],
+ "page_idx": 7
+ },
+ {
+ "type": "header",
+ "text": "J. Zheng et al.",
+ "bbox": [
+ 271,
+ 114,
+ 369,
+ 128
+ ],
+ "page_idx": 7
+ },
+ {
+ "type": "text",
+ "text": "Theorem 1. For the sequence $\\{[\\pmb{\\mathcal{Z}}^{(t)},\\{\\pmb{U}_{k_n}^{(t)}\\}_{n = s + 1}^h,\\pmb{\\mathcal{E}}^{(t)},\\pmb{\\mathcal{V}}^{(t)},\\pmb{\\mu}^{(t)}]\\}$ generated by the proposed algorithm 1, we have the following properties if $\\{\\pmb{\\mathcal{V}}^{(t)}\\}$ is bounded, $\\sum_{t = 1}^{\\infty}(\\mu^{(t)})^{-2}\\mu^{(t + 1)} < + \\infty$ and $\\lim_{n\\to \\infty}\\mu^{(n)}\\sum_{t = n}^{\\infty}(\\eta^{(t)})^{-1 / 2} = 0.$",
+ "bbox": [
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+ "page_idx": 8
+ },
+ {
+ "type": "text",
+ "text": "(i) $\\lim_{t\\to \\infty}\\Psi_{\\mathbb{I}}(\\boldsymbol {M}) - \\boldsymbol{x}^{(t)} - \\boldsymbol{\\varepsilon}^{(t)} = \\mathbf{0},$ where $\\pmb {\\mathcal{X}}^{(t)} = \\pmb {\\mathcal{Z}}^{(t)}\\times_{k_h}(\\pmb{U}_{k_h}^{(t)})^T\\dots \\times_{k_{s + 1}}$ $(\\pmb{U}_{k_{s + 1}}^{(t)})^T$",
+ "bbox": [
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+ ],
+ "page_idx": 8
+ },
+ {
+ "type": "text",
+ "text": "(ii) $\\{[\\pmb{\\mathcal{Z}}^{(t)},\\{\\pmb{U}_{k_n}^{(t)}\\}_{n = s + 1}^h ],\\pmb{\\mathcal{X}}^{(t)},\\pmb{\\mathcal{E}}^{(t)}]\\}$ is bounded.",
+ "bbox": [
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+ "page_idx": 8
+ },
+ {
+ "type": "text",
+ "text": "(iii) $\\sum_{t=1}^{\\infty} \\eta^{(t)} \\| [\\pmb{\\mathcal{Z}}^{(t)}, \\{\\pmb{U}_{k_n}^{(t)}\\}_{n=s+1}^h, \\pmb{\\mathcal{E}}^{(t)}] - [\\pmb{\\mathcal{Z}}^{(t+1)}, \\{\\pmb{U}_{k_n}^{(t+1)}\\}_{n=s+1}^h, \\pmb{\\mathcal{E}}^{(t+1)}] \\|_F^2$ is convergent. Thus, we have",
+ "bbox": [
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+ "page_idx": 8
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\| [ \\boldsymbol {\\mathcal {Z}} ^ {(t)}, \\{\\boldsymbol {U} _ {k _ {n}} ^ {(t)} \\} _ {n = s + 1} ^ {h}, \\boldsymbol {\\mathcal {E}} ^ {(t)} ] - [ \\boldsymbol {\\mathcal {Z}} ^ {(t + 1)}, \\{\\boldsymbol {U} _ {k _ {n}} ^ {(t + 1)} \\} _ {n = s + 1} ^ {h}, \\boldsymbol {\\mathcal {E}} ^ {(t + 1)} ] \\| _ {F} ^ {2} \\leq \\mathcal {O} (\\frac {1}{\\eta^ {(t)}}).\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ {
+ "type": "list",
+ "sub_type": "text",
+ "list_items": [
+ "$(iv)$ $\\lim_{t\\to \\infty}\\| \\pmb{\\mathcal{V}}^{(t + 1)} - \\pmb{\\mathcal{V}}^{(t)}\\| _F = 0.$",
+ "(v) Let $[\\mathcal{Z}^*,\\{\\pmb{U}_{k_n}^*\\}_{n = s + 1},\\pmb {\\varepsilon}^*,\\pmb {\\mathcal{V}}^* ]$ be any limit point of $\\{[\\pmb {\\mathcal{Z}}^{(t)},\\{\\pmb{U}_{k_n}^{(t)}\\}_{n = s + 1}^h,\\pmb {\\mathcal{E}}^{(t)},$ $\\pmb {\\mathcal{V}}^{(t)}]\\}$ . Then, $[\\pmb {\\mathcal{Z}}^*,\\{\\pmb{U}_{k_n}^*\\}_{n = s + 1}^h,\\pmb {\\mathcal{E}}^*,\\pmb {\\mathcal{V}}^* ]$ is a KKT point to (8)."
+ ],
+ "bbox": [
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+ ],
+ "page_idx": 8
+ },
+ {
+ "type": "text",
+ "text": "Please refer to the supplementary material of this paper for the proof of the Theorem 1. Theorem 1 shows that, if $\\{\\pmb{v}^{(t)}\\}$ is bounded, the sequence $[\\pmb{z}^{(t)},\\{\\pmb{U}_{k_n}^{(t)}\\}_{n = s + 1}^h,\\pmb{\\varepsilon}^{(t)}]$ generated by the proposed algorithm 1 is Cauchy convergent, with a convergence rate of at least $\\mathcal{O}\\left(\\frac{1}{\\eta^{(t)}}\\right)$ . Moreover, any accumulation point of the sequence converges to the KKT point of (8).",
+ "bbox": [
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+ "page_idx": 8
+ },
+ {
+ "type": "text",
+ "text": "3.2 APMM-based Heuristic Method for Solving MOTC",
+ "text_level": 1,
+ "bbox": [
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+ ],
+ "page_idx": 8
+ },
+ {
+ "type": "text",
+ "text": "To solve MOTC, we iteratively update $U_{k_n}^{(k_1,k_2)}(n = 1 + s,\\dots ,h,1\\leq k_1 < k_2\\leq h)$ and $\\mathcal{X}$ in (7) by solving the following optimization problems:",
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+ "page_idx": 8
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\hat {\\mathcal {U}} _ {(k _ {1}, k _ {2})} = \\underset {\\tilde {\\mathcal {U}} _ {(k _ {1}, k _ {2})}} {\\arg \\min } \\| \\hat {\\boldsymbol {X}} \\| _ {*}, \\tilde {\\mathcal {U}} _ {(k _ {1}, k _ {2})} ^ {[ k _ {1}, k _ {2} ]} \\tag {10}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 8
+ },
+ {
+ "type": "text",
+ "text": "and",
+ "bbox": [
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+ "page_idx": 8
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\hat {\\boldsymbol {\\mathcal {X}}} = \\underset {\\boldsymbol {\\mathcal {X}}} {\\arg \\min } \\left[ \\| \\boldsymbol {\\mathcal {X}} \\| _ {*} ^ {(k _ {1}, k _ {2})} \\right] _ {1 \\leq k _ {1} < k _ {2} \\leq h} s. t. \\Psi_ {\\mathbb {I}} (\\boldsymbol {\\mathcal {M}}) = \\Psi_ {\\mathbb {I}} (\\boldsymbol {\\mathcal {X}}), \\tag {11}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 8
+ },
+ {
+ "type": "text",
+ "text": "respectively. We use the proposed APMM for solving (10), and the non-dominated sorting genetic algorithm (NSGA-II) [5] presented in Algorithm 2 for solving (11). These two steps are used for the learning of the tensor nuclear norm functions and the low rank estimation, respectively. It is worth noting that both updating the $\\hat{\\mathcal{U}}_{(k_1,k_2)}$ for each $(k_{1},k_{2})$ and evaluating the individuals in Algorithm 2 can be proceed in parallel, which can accelerate the whole optimization for MOTC.",
+ "bbox": [
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+ ],
+ "page_idx": 8
+ },
+ {
+ "type": "header",
+ "text": "Handling The Non-Smooth Challenge in Tensor SVD",
+ "bbox": [
+ 375,
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+ ],
+ "page_idx": 8
+ },
+ {
+ "type": "page_number",
+ "text": "9",
+ "bbox": [
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+ "page_idx": 8
+ },
+ {
+ "type": "code",
+ "sub_type": "algorithm",
+ "code_caption": [
+ "Algorithm 2: NSGA-II-based framework for solving (11)"
+ ],
+ "code_body": "Input: $\\Psi_{\\mathbb{I}}(\\mathcal{M}),\\{\\hat{\\mathcal{U}}_{(k_1,k_2)}\\}_{1\\leq k_1 < k_2\\leq h}$ , IterMax, and $t = 0$ \nOutput: $\\hat{\\pmb{x}}$ \n1. Initialize: population $\\mathbb{P}^{(0)}$ \n2. Evaluate $\\left[\\| \\pmb {x}\\|_{*,\\hat{\\pmb{u}}_{(k_1,k_2)}}^{[k_1,k_2]}\\right]_{1\\leq k_1 < k_2\\leq h}$ for each $\\pmb {\\chi}\\in \\mathbb{P}^{(t)}$ . \n3. Sorting individuals by their non-domination ranks and crowding distance, and obtaining the first front $\\mathbb{F}_1$ . \n4. Selecting parents and applying crossover and mutation to create offspring; \n5. Obtaining the next generation by truncating the sorted individuals (including offspring population) and update the first front $\\mathbb{F}_1$ . \n6. $t = t + 1$ \n7. Repeat steps 2-6 until $t = \\mathrm{IterMax}$ \n8. $\\hat{\\pmb{x}} =$ average sum of the individuals that belong to $\\mathbb{F}_1$",
+ "bbox": [
+ 241,
+ 170,
+ 759,
+ 348
+ ],
+ "page_idx": 9
+ },
+ {
+ "type": "text",
+ "text": "4 Experimental Results",
+ "text_level": 1,
+ "bbox": [
+ 215,
+ 377,
+ 460,
+ 395
+ ],
+ "page_idx": 9
+ },
+ {
+ "type": "text",
+ "text": "In this section, we compared the TC-SL and MOTC| Notations | Descriptions | Notations | Descriptions |
| a, b, c, ... | calvars | a, b, c, ... | vectors |
| A, B, C, ... | matrices | A, B, C, ... | tensors |
| A, B, C, ... | sets | A, B, C, ... | operators |
| [1:n] | {1,2,...,n} | AT | transpose of A |
| I | identity tensor | 0 | null tensor |
| [At1,i2,...,ih] | (i1,i2,...,ih)-th element of A | ||A||0 | the number of non-zero elements of A |
| ||A||1 | ||A||1 = ∑i1,i2,...,ih ||A||i1,i2,...,ih| | ATk1,...,kh | rotation of A ∈ R1×...×Ih such that ATk1,...,kh ∈ Rl2×...×Ikh |
| [Atk1,...,kh], ...,ik3,...,ikh | slice along the (k1,k2)-th mode | A(k) (or [Aj(n)]) | Mode-n Unfolding of A |
| ×n | Mode-n product | Ac | complementary set of A |
"
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.216,
+ 0.338,
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+ 0.353
+ ],
+ "angle": 0,
+ "content": "2.1 Tensor Completion Problem"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.214,
+ 0.361,
+ 0.785,
+ 0.392
+ ],
+ "angle": 0,
+ "content": "Considering tensor data \\(\\mathcal{M} \\in \\mathbb{R}^{I_1 \\times I_2 \\times \\dots \\times I_h}\\) may with missing elements, [17, 20] propose the following t-SVD-based tensor completion model:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.348,
+ 0.402,
+ 0.788,
+ 0.425
+ ],
+ "angle": 0,
+ "content": "\\[\n\\min _ {\\boldsymbol {x}} \\operatorname {r a n k} _ {[ k _ {1}, k _ {2} ]} (\\boldsymbol {x}) \\quad s. t. \\Psi_ {\\mathbb {I}} (\\boldsymbol {\\mathcal {M}}) = \\Psi_ {\\mathbb {I}} (\\boldsymbol {x}), \\tag {1}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.214,
+ 0.434,
+ 0.7,
+ 0.45
+ ],
+ "angle": 0,
+ "content": "where \\(\\varPsi_{\\mathbb{I}}\\) is a linear project operator on the support set \\(\\mathbb{I}\\) such that"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.278,
+ 0.458,
+ 0.721,
+ 0.492
+ ],
+ "angle": 0,
+ "content": "\\[\n[ \\varPsi_{\\mathbb{I}}(\\mathcal{M})]_{i_{1},i_{2},\\dots ,i_{h}}=\\left\\{\\begin{array}{ll}[\\mathcal{M}]_{i_{1},i_{2},\\cdots,i_{h}},& \\mbox{if } (i_{1},i_{2},\\cdots,i_{h})\\in\\mathbb{I};\\\\ 0,&\\mbox{if }(i_{1},i_{2},\\cdots,i_{h})\\notin\\mathbb{I},\\end{array}\\right.\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.214,
+ 0.501,
+ 0.787,
+ 0.547
+ ],
+ "angle": 0,
+ "content": "and \\( i_k \\in [1: I_k] \\) for \\( k = 1, 2, \\dots, h \\). The tensor rank function in (1) operates on the assumption that an \\( h \\)-order tensor data \\( \\mathcal{M} \\in \\mathbb{R}^{I_1 \\times I_2 \\times \\dots \\times I_h} \\) in the real world can be decomposed as"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.391,
+ 0.555,
+ 0.788,
+ 0.58
+ ],
+ "angle": 0,
+ "content": "\\[\n\\boldsymbol {\\mathcal {M}} = \\boldsymbol {\\mathcal {Z}} \\times_ {k _ {3}} \\hat {\\boldsymbol {U}} _ {k _ {3}} ^ {T} \\dots \\times_ {k _ {h}} \\hat {\\boldsymbol {U}} _ {k _ {h}} ^ {T}, \\tag {2}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.214,
+ 0.589,
+ 0.788,
+ 0.62
+ ],
+ "angle": 0,
+ "content": "where \\(\\{k_3, k_4, \\dots, k_h\\} \\subset [1:h]\\), \\(\\{\\hat{U}_{k_n}\\}_{n=3}^h\\) are a set of given invertible transforms, and"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.251,
+ 0.627,
+ 0.787,
+ 0.677
+ ],
+ "angle": 0,
+ "content": "\\[\n\\begin{array}{l} \\operatorname {r a n k} _ {[ k _ {1}, k _ {2} ]} (\\boldsymbol {\\mathcal {M}}) = \\max _ {i _ {k _ {n}} \\in [ 1: I _ {k _ {n}} ] \\text {f o r} n = 3, 4, \\dots , h} \\operatorname {r a n k} \\left(\\left[ \\boldsymbol {\\mathcal {Z}} ^ {T _ {k _ {1}}, \\dots , k _ {h}} \\right] _ {:,: i _ {k _ {3}}, \\dots , i _ {k _ {h}}}\\right) \\\\ < < \\min \\left(I _ {k _ {1}}, I _ {k _ {2}}\\right) \\tag {3} \\\\ \\end{array}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.214,
+ 0.688,
+ 0.787,
+ 0.722
+ ],
+ "angle": 0,
+ "content": "for certain \\((k_{1},k_{2})\\) satisfying \\(1\\leq k_{1} < k_{2}\\leq h\\). If taken \\(\\{\\hat{U}_{k_n}\\}_{n = 3}^h\\) as Discrete Fourier Matrices, \\(\\mathrm{rank}_{[k_1,k_2]}(\\mathcal{M})\\) is referred as to tensor tubal rank of \\(\\mathcal{M}\\) [16]."
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.214,
+ 0.74,
+ 0.764,
+ 0.77
+ ],
+ "angle": 0,
+ "content": "2.2 The Proposed Tensor Completion with A Learnable Tensor Rank Function"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.214,
+ 0.78,
+ 0.787,
+ 0.841
+ ],
+ "angle": 0,
+ "content": "From the definition of \\(\\mathrm{rank}_{[k_1,k_2]}(\\mathcal{M})\\), we can see that it aims to examine the slice-wise tensor rank along the \\((k_{1},k_{2})\\)-th mode of transformed data \\(\\pmb{\\mathcal{Z}}\\). It allows (1) to investigate the low-rankness of different features in tensor data corresponding to different frequencies separately and jointly."
+ }
+ ],
+ [
+ {
+ "type": "header",
+ "bbox": [
+ 0.377,
+ 0.115,
+ 0.733,
+ 0.129
+ ],
+ "angle": 0,
+ "content": "Handling The Non-Smooth Challenge in Tensor SVD"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.775,
+ 0.117,
+ 0.785,
+ 0.127
+ ],
+ "angle": 0,
+ "content": "5"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.215,
+ 0.147,
+ 0.788,
+ 0.207
+ ],
+ "angle": 0,
+ "content": "However, when \\(\\mathcal{M}\\) exhibits harsh changes caused by disordered tensor slices along a certain mode \\(k_{n}\\), all information in \\(\\mathcal{M}\\) tends to collapse into the high-frequency slices in \\(\\mathcal{Z}\\). To handle such situation, we introduce a learnable permutation matrix \\(P\\in \\mathbb{R}^{I_{k_n}\\times I_{k_n}}\\) to (2):"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.284,
+ 0.218,
+ 0.721,
+ 0.24
+ ],
+ "angle": 0,
+ "content": "\\[\n\\pmb {\\mathcal {M}} \\times_ {k _ {n}} \\pmb {P} = \\pmb {\\mathcal {Z}} \\times_ {k _ {3}} \\hat {\\pmb {U}} _ {k _ {3}} ^ {T} \\dots \\times_ {k _ {n}} (\\hat {\\pmb {U}} _ {k _ {n}} \\pmb {P}) ^ {T} \\times_ {k _ {n + 1}} \\dots \\times_ {k _ {h}} \\hat {\\pmb {U}} _ {k _ {h}} ^ {T},\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.216,
+ 0.248,
+ 0.547,
+ 0.263
+ ],
+ "angle": 0,
+ "content": "and therefore we can get the following model:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.287,
+ 0.273,
+ 0.787,
+ 0.364
+ ],
+ "angle": 0,
+ "content": "\\[\n\\begin{array}{l} \\min _ {\\boldsymbol {X}, \\boldsymbol {P} _ {k _ {n}}} \\operatorname {r a n k} _ {[ k _ {1}, k _ {2} ]} (\\boldsymbol {X} \\times_ {k _ {n}} \\boldsymbol {P} _ {k _ {n}}) \\\\ s. t. \\Psi_ {\\mathbb {I}} (\\boldsymbol {\\mathcal {M}}) = \\Psi_ {\\mathbb {I}} (\\boldsymbol {\\mathcal {X}}), \\\\ \\sum_ {i = 1} ^ {I _ {k _ {n}}} \\left[ \\boldsymbol {P} _ {k _ {n}} \\right] _ {i, j} = 1, \\quad \\sum_ {j = 1} ^ {I _ {k _ {n}}} \\left[ \\boldsymbol {P} _ {k _ {n}} \\right] _ {i, j} = 1 \\text {f o r} \\left[ \\boldsymbol {P} _ {k _ {n}} \\right] _ {i, j} \\in \\{0, 1 \\}. \\tag {4} \\\\ \\end{array}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.214,
+ 0.373,
+ 0.786,
+ 0.417
+ ],
+ "angle": 0,
+ "content": "This incorporation is aimed at addressing the slice permutation property in tensor completion methods, thus facilitating a more effective exploration of the low-rank property in \\(\\mathcal{M}\\)."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.214,
+ 0.418,
+ 0.8,
+ 0.485
+ ],
+ "angle": 0,
+ "content": "Unfortunately, solving (4) is challenging due to the constraints of \\(\\sum_{i=1}^{I_{k_n}}[P_{k_n}]_{i,j} = 1\\) and \\(\\sum_{j=1}^{I_{k_n}}[P_{k_n}]_{i,j} = 1\\) for \\([P_{k_n}]_{i,j} \\in \\{0,1\\}\\). Therefore, we opt to use a set of learnable unitary matrices \\(\\{\\pmb{U}_{k_n}\\}_{n=s+1}^h\\) instead, proposing the following tensor completion model for given \\(\\{k_{s+1}, k_{s+2}, \\dots, k_h\\}\\):"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.262,
+ 0.496,
+ 0.787,
+ 0.541
+ ],
+ "angle": 0,
+ "content": "\\[\n\\begin{array}{l} \\min _ {\\boldsymbol {X}, \\boldsymbol {U} _ {k _ {n}} (n = s + 1, \\dots , h)} \\operatorname {r a n k} _ {[ k _ {1}, k _ {2} ]} (\\boldsymbol {\\mathcal {X}} \\times_ {k _ {s + 1}} \\boldsymbol {U} _ {k _ {s + 1}} \\dots \\times_ {k _ {h}} \\boldsymbol {U} _ {k _ {h}}) \\\\ s. t. \\Psi_ {\\mathbb {I}} (\\boldsymbol {\\mathcal {M}}) = \\Psi_ {\\mathbb {I}} (\\boldsymbol {\\mathcal {X}}), \\boldsymbol {U} _ {k _ {n}} ^ {T} \\boldsymbol {U} _ {k _ {n}} = \\boldsymbol {I} (n = s + 1, \\dots , h), \\tag {5} \\\\ \\end{array}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.215,
+ 0.55,
+ 0.788,
+ 0.653
+ ],
+ "angle": 0,
+ "content": "where \\(\\mathcal{X}\\) is a low-rank estimation of the true tensor data \\(\\mathcal{M}\\). Besides simplifying the optimization process of (4), the introduction of the learnable unitary matrices enables the model to extract features for better studying the low-rankness in tensor data and to handle other scenarios where the initially provided transforms \\(\\{\\hat{U}_{k_n}\\}_{n=3}^h\\) are inadequate, such as videos with irregularly changing content and image sequences with different scenes. Hence, we introduce our learnable tensor rank as"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.368,
+ 0.655,
+ 0.635,
+ 0.674
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathrm {r a n k} _ {[ k _ {1}, k _ {2} ], \\tilde {\\mathcal {U}}} (\\pmb {\\mathscr {X}}) = \\mathrm {r a n k} _ {[ k _ {1}, k _ {2} ]} (\\tilde {\\mathcal {U}} (\\pmb {\\mathscr {X}})),\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.215,
+ 0.679,
+ 0.528,
+ 0.697
+ ],
+ "angle": 0,
+ "content": "where \\(\\tilde{\\mathcal{U}} (\\pmb {x}) = \\pmb {x}\\times_{k_{s + 1}}\\pmb{U}_{k_{s + 1}}\\dots \\times_{k_h}\\pmb{U}_{k_h}.\\)"
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.215,
+ 0.717,
+ 0.787,
+ 0.748
+ ],
+ "angle": 0,
+ "content": "2.3 Approximation to The Proposed Tensor Completion by Using A Learnable Tensor Nuclear Norm (TC-SL)"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.215,
+ 0.755,
+ 0.788,
+ 0.841
+ ],
+ "angle": 0,
+ "content": "Since the function \\(\\mathrm{rank}_{[k_1,k_2],\\hat{\\mathcal{U}}}(\\cdot)\\) is discrete, it often leads to the NP-hard problem. From the result given in [16,17], we know that, when \\(h = 3\\) and \\(\\hat{U}_3\\) is orthogonal, \\(\\| \\cdot \\|_{*}^{[1,2]}\\) is the tightest convex envelope of \\(\\mathrm{rank}_{[1,2]}(\\cdot)\\) on the set \\(\\{\\mathcal{A}||\\mathcal{A}||_2^{[1,2]}\\leq 1\\}\\), where the definitions of \\(\\| \\cdot \\| _2^{[k_1,k_2]}\\) and \\(\\| \\cdot \\|_{*}^{[k_{1},k_{2}]}\\) are given in Definition 1. This conclusion can be easily extended to our case, and we obtain the Property 1."
+ }
+ ],
+ [
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.218,
+ 0.116,
+ 0.23,
+ 0.127
+ ],
+ "angle": 0,
+ "content": "6"
+ },
+ {
+ "type": "header",
+ "bbox": [
+ 0.272,
+ 0.115,
+ 0.37,
+ 0.129
+ ],
+ "angle": 0,
+ "content": "J. Zheng et al."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.214,
+ 0.144,
+ 0.79,
+ 0.178
+ ],
+ "angle": 0,
+ "content": "Definition 1. For an \\( h \\)-order tensor \\( \\mathcal{A} \\in \\mathbb{R}^{I_1 \\times I_2 \\times \\dots \\times I_h} \\), \\( \\| \\mathcal{A} \\|_*^{[k_1, k_2]} \\) and \\( \\| \\mathcal{A} \\|_2^{[k_1, k_2]} \\) are defined as"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.248,
+ 0.187,
+ 0.756,
+ 0.224
+ ],
+ "angle": 0,
+ "content": "\\[\n\\| \\boldsymbol {\\mathcal {A}} \\| _ {*} ^ {[ k _ {1}, k _ {2} ]} = \\sum_ {i _ {k _ {3}}, i _ {k _ {4}}, \\dots , i _ {k _ {h}}} \\| [ \\boldsymbol {\\mathcal {A}} ^ {T _ {k _ {1}}, \\dots , k _ {h}} \\times_ {k _ {3}} \\hat {\\boldsymbol {U}} _ {k _ {3}} \\cdot \\cdot \\cdot \\times_ {k _ {h}} \\hat {\\boldsymbol {U}} _ {k _ {h}} ] _ {:;,: i _ {k _ {3}}, \\dots , i _ {k _ {h}}} \\| _ {*}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.216,
+ 0.234,
+ 0.248,
+ 0.247
+ ],
+ "angle": 0,
+ "content": "and"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.246,
+ 0.257,
+ 0.758,
+ 0.286
+ ],
+ "angle": 0,
+ "content": "\\[\n\\| \\pmb {\\mathcal {A}} \\| _ {2} ^ {[ k _ {1}, k _ {2} ]} = \\max _ {i _ {k _ {3}}, i _ {k _ {4}}, \\dots , i _ {k _ {h}}} \\| [ \\pmb {\\mathcal {A}} ^ {T _ {k _ {1}}, \\dots , k _ {h}} \\times_ {k _ {3}} \\hat {\\pmb {U}} _ {k _ {3}} \\cdot \\cdot \\cdot \\times_ {k _ {h}} \\hat {\\pmb {U}} _ {k _ {h}} ] _ {:,: i _ {k _ {3}}, \\dots , i _ {k _ {h}}} \\| _ {2},\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.216,
+ 0.296,
+ 0.304,
+ 0.312
+ ],
+ "angle": 0,
+ "content": "respectively."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.216,
+ 0.321,
+ 0.798,
+ 0.41
+ ],
+ "angle": 0,
+ "content": "Property 1. For an h-order tensor \\(\\mathcal{A} \\in \\mathbb{R}^{I_1 \\times I_2 \\times \\dots \\times I_h}\\), let us define \\(\\| \\mathcal{A} \\|_{*,\\tilde{\\mathcal{U}}}^{[k_1,k_2]}\\) and \\(\\| \\mathcal{A} \\|_{2,\\tilde{\\mathcal{U}}}^{[k_1,k_2]}\\) are defined as \\(\\| \\mathcal{A} \\|_{*,\\tilde{\\mathcal{U}}}^{[k_1,k_2]} = \\| \\tilde{\\mathcal{U}} (\\mathcal{A}) \\|_{*}^{[k_1,k_2]}\\) and \\(\\| \\mathcal{A} \\|_{2,\\tilde{\\mathcal{U}}}^{[k_1,k_2]} = \\| \\tilde{\\mathcal{U}} (\\mathcal{A}) \\|_{2}^{[k_1,k_2]}\\), respectively. \\(\\| \\cdot \\|_{*,\\tilde{\\mathcal{U}}}^{[k_1,k_2]}\\) is the dual norm of tensor \\(\\| \\cdot \\|_{2,\\tilde{\\mathcal{U}}}^{[k_1,k_2]}\\) norm, and \\(\\| \\cdot \\|_{*,\\tilde{\\mathcal{U}}}^{[k_1,k_2]}\\) is the tightest convex envelope of \\(\\mathrm{rank}_{[k_1,k_2],\\tilde{\\mathcal{U}}}(\\cdot)\\) on the set \\(\\{\\mathcal{A} \\| \\mathcal{A} \\|_{2,\\tilde{\\mathcal{U}}}^{[k_1,k_2]} \\leq 1\\}\\)."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.214,
+ 0.419,
+ 0.788,
+ 0.465
+ ],
+ "angle": 0,
+ "content": "Therefore, we derive an approximation to the proposed model (6) by utilizing a learnable tensor nuclear norm based on the prior assumption of slice-wise low-rankness in the transformed data (TC-SL):"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.357,
+ 0.475,
+ 0.788,
+ 0.504
+ ],
+ "angle": 0,
+ "content": "\\[\n\\min _ {\\boldsymbol {X}, \\tilde {\\mathcal {U}}} \\| \\boldsymbol {X} \\| _ {*}, \\tilde {\\mathcal {U}} ^ {[ k _ {1}, k _ {2} ]} \\quad s. t. \\Psi_ {\\mathbb {I}} (\\boldsymbol {\\mathcal {M}}) = \\Psi_ {\\mathbb {I}} (\\boldsymbol {\\mathcal {X}}). \\tag {6}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.214,
+ 0.517,
+ 0.788,
+ 0.566
+ ],
+ "angle": 0,
+ "content": "Let \\(\\hat{\\mathcal{U}}\\) be the invertible transform operator learned by (6). The exactly recovery of TC-SL with given \\(\\hat{\\mathcal{U}}\\) is guaranteed from the current studies in the exactly recovery of the t-SVD-based tensor completion [16, 17]."
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.216,
+ 0.585,
+ 0.787,
+ 0.616
+ ],
+ "angle": 0,
+ "content": "2.4 The Proposed Multi-Objective Tensor Completion to Learn The Cross-Dimensional Low-Rankness"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.214,
+ 0.625,
+ 0.788,
+ 0.717
+ ],
+ "angle": 0,
+ "content": "From (6), we observe that the definition of TC-SL depends on the choice of \\( k_{1} \\) and \\( k_{2} \\), and it considers different kinds of low-rankness in the tensor data by adjusting \\( (k_{1}, k_{2}) \\). However, considering only one mode may result in the loss of correlation information across the remaining modes. To address this issue, we give the following multi-objective model with learnable tensor nuclear norms for tensor completion (MOTC):"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.287,
+ 0.727,
+ 0.788,
+ 0.761
+ ],
+ "angle": 0,
+ "content": "\\[\n\\min _ {\\boldsymbol {x}, \\tilde {\\mathcal {U}} _ {\\left(k _ {1}, k _ {2}\\right)}} \\left[ \\| \\boldsymbol {\\mathcal {X}} \\| _ {*}, \\tilde {\\mathcal {U}} _ {\\left(k _ {1}, k _ {2}\\right)} ^ {[ k _ {1}, k _ {2} ]} \\right] _ {1 \\leq k _ {1} < k _ {2} \\leq h} \\quad s. t. \\Psi_ {\\mathbb {I}} (\\boldsymbol {\\mathcal {M}}) = \\Psi_ {\\mathbb {I}} (\\boldsymbol {\\mathcal {X}}), \\tag {7}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.214,
+ 0.77,
+ 0.788,
+ 0.84
+ ],
+ "angle": 0,
+ "content": "where \\(\\tilde{\\mathcal{U}}_{(k_1,k_2)}(\\boldsymbol {x}) = \\boldsymbol {x}\\times_{k_{s + 1}}\\boldsymbol{U}_{k_{s + 1}}^{(k_1,k_2)}\\dots \\times_{k_h}\\boldsymbol{U}_{k_h}^{(k_1,k_2)}\\) and \\((\\boldsymbol{U}_{k_n}^{(k_1,k_2)})^T\\boldsymbol{U}_{k_n}^{(k_1,k_2)} = \\boldsymbol{I}\\) for \\(n = 1 + s,\\dots ,h\\) and \\(1\\leq k_{1} < k_{2}\\leq h\\). In MOTC, the multiple objective functions \\(\\| \\pmb {x}\\|_{*,\\tilde{\\mathcal{U}}_{(k_1,k_2)}}^{[k_1,k_2]}\\) \\((1\\leq k_{1} < k_{2}\\leq h)\\) are utilized to examine the low-rankness of tensor data from its various dimensions."
+ }
+ ],
+ [
+ {
+ "type": "header",
+ "bbox": [
+ 0.377,
+ 0.115,
+ 0.733,
+ 0.129
+ ],
+ "angle": 0,
+ "content": "Handling The Non-Smooth Challenge in Tensor SVD"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.775,
+ 0.116,
+ 0.785,
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+ ],
+ "angle": 0,
+ "content": "7"
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.216,
+ 0.145,
+ 0.486,
+ 0.163
+ ],
+ "angle": 0,
+ "content": "3 Optimization Algorithm"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ 0.193
+ ],
+ "angle": 0,
+ "content": "In this section, we provide a detailed discussion for the optimizing (6) and (7)."
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.215,
+ 0.212,
+ 0.773,
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+ ],
+ "angle": 0,
+ "content": "3.1 Alternating Proximal Multiplier Method (APMM) for TC-SL"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.214,
+ 0.237,
+ 0.787,
+ 0.283
+ ],
+ "angle": 0,
+ "content": "To solve the problem (6), we introduce auxiliary variables \\(\\pmb{\\mathcal{E}}\\in \\mathbb{E} = \\{\\pmb {\\mathcal{E}}|\\Psi_{\\mathbb{I}}(\\pmb {\\mathcal{E}}) = \\mathbf{0}\\}\\) and \\(\\pmb{\\Xi}\\) such that \\(\\pmb {x} = \\pmb {z}\\times_{k_{s + 1}}\\pmb{U}_{k_{s + 1}}^T\\dots \\times_{k_h}\\pmb{U}_{k_h}^T\\) . Therefore, we turn to solve the following equivalence problem:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.217,
+ 0.291,
+ 0.796,
+ 0.337
+ ],
+ "angle": 0,
+ "content": "\\[\n\\min _ {\\boldsymbol {Z}, \\boldsymbol {U} _ {k _ {n}} ^ {T} \\boldsymbol {U} _ {k _ {n}} = \\boldsymbol {I} (n = s + 1, \\dots , h)} \\| \\boldsymbol {Z} \\| _ {*} ^ {[ k _ {1}, k _ {2} ]} \\quad \\text {s . t .} \\Psi_ {\\mathbb {I}} (\\boldsymbol {\\mathcal {M}}) = \\boldsymbol {\\mathcal {Z}} \\times_ {k _ {s + 1}} \\boldsymbol {U} _ {k _ {s + 1}} ^ {T} \\dots \\times_ {k _ {h}} \\boldsymbol {U} _ {k _ {h}} ^ {T} + \\boldsymbol {\\mathcal {E}}. \\tag {8}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.215,
+ 0.337,
+ 0.651,
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+ ],
+ "angle": 0,
+ "content": "The Augmented Lagrangian function of (8) is formulated as"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.217,
+ 0.361,
+ 0.79,
+ 0.412
+ ],
+ "angle": 0,
+ "content": "\\[\n\\begin{array}{l} \\mathcal {L} (\\boldsymbol {\\mathcal {Z}}, \\{\\boldsymbol {U} _ {k _ {n}} \\} _ {n = s + 1} ^ {h}, \\boldsymbol {\\mathcal {E}}, \\boldsymbol {\\mathcal {Y}}, \\mu) = \\| \\boldsymbol {\\mathcal {Z}} \\| _ {*} ^ {[ k _ {1}, k _ {2} ]} + \\left\\langle \\Psi_ {\\mathbb {I}} (\\boldsymbol {\\mathcal {M}}) - \\boldsymbol {\\mathcal {Z}} \\times_ {k _ {s + 1}} \\boldsymbol {U} _ {k _ {s + 1}} ^ {T} \\dots \\times_ {k _ {h}} \\boldsymbol {U} _ {k _ {h}} ^ {T} \\right. \\\\ - \\mathcal {E}, \\mathcal {Y} \\rangle + \\frac {\\mu}{2} \\| \\Psi_ {\\mathbb {I}} (\\mathcal {M}) - \\mathcal {Z} \\times_ {k _ {s + 1}} \\boldsymbol {U} _ {k _ {s + 1}} ^ {T} \\dots \\times_ {k _ {h}} \\boldsymbol {U} _ {k _ {h}} ^ {T} - \\mathcal {E} \\| _ {F} ^ {2}, \\tag {9} \\\\ \\end{array}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.215,
+ 0.42,
+ 0.788,
+ 0.485
+ ],
+ "angle": 0,
+ "content": "where \\(\\mathcal{V}\\) is Lagrange multiplier, and \\(\\mu\\) is a positive scalar. We solve (8) iteratively by combining the proximal algorithm with the Alternating Direction Method of Multipliers (APMM) that is given in the Algorithm 1. We detail the solutions for solving \\(\\mathcal{Z}^{(t + 1)}, U_{k_n}^{(t + 1)}\\), and \\(\\mathcal{E}^{(t + 1)}\\) as follows."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.227,
+ 0.505,
+ 0.384,
+ 0.521
+ ],
+ "angle": 0,
+ "content": "Calculate \\(\\mathcal{Z}^{(t + 1)}\\)"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.255,
+ 0.529,
+ 0.776,
+ 0.602
+ ],
+ "angle": 0,
+ "content": "\\[\n\\begin{array}{l} \\boldsymbol {\\mathcal {Z}} ^ {(t + 1)} = \\underset {\\boldsymbol {\\mathcal {Z}}} {\\arg \\min } \\frac {1}{2} \\| \\frac {\\mu^ {(t)} \\hat {\\boldsymbol {\\mathcal {P}}} \\times_ {k _ {s + 1}} \\boldsymbol {U} _ {k _ {s + 1}} ^ {(t)} \\cdots \\times_ {k _ {h}} \\boldsymbol {U} _ {k _ {h}} ^ {(t)} + \\eta^ {(t)} \\boldsymbol {\\mathcal {Z}} ^ {(t)}}{\\mu^ {(t)} + \\eta^ {(t)}} - \\boldsymbol {\\mathcal {Z}} \\| _ {F} ^ {2} \\\\ + \\frac {1}{\\mu^ {(t)} + \\eta^ {(t)}} \\| \\mathcal {Z} \\| _ {*} ^ {[ k _ {1}, k _ {2} ]}, \\\\ \\end{array}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.241,
+ 0.611,
+ 0.787,
+ 0.671
+ ],
+ "angle": 0,
+ "content": "where \\(\\hat{\\mathcal{P}} = \\Psi_{\\mathbb{I}}(\\mathcal{M}) - \\pmb{\\mathcal{E}}^{(t)} + \\frac{1}{\\mu^{(t)}}\\pmb{\\mathcal{Y}}^{(t)}\\). It can be solved by the tensor singular value thresholding operation with parameter \\(\\frac{1}{\\mu^{(t)} + \\eta^{(t)}}\\) [15]."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.227,
+ 0.673,
+ 0.788,
+ 0.715
+ ],
+ "angle": 0,
+ "content": "Calculate \\(U_{k_n}^{(t + 1)}\\): Let \\(\\mathcal{A} = \\mathcal{Z}^{(t + 1)} \\times_{k_h} U_{k_h}^{(t)T} \\times_{k_h - 1} \\dots \\times_{k_n + 1} U_{k_n + 1}^{(t)T}\\) and \\(\\mathcal{B} = \\hat{\\mathcal{P}} \\times_{s + 1} U_{k_{s + 1}}^{(t + 1)} \\dots \\times_{k_n - 1} U_{k_n - 1}^{(t + 1)}\\), and we have"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.243,
+ 0.724,
+ 0.785,
+ 0.757
+ ],
+ "angle": 0,
+ "content": "\\[\n\\boldsymbol {U} _ {k _ {n}} ^ {(t + 1)} = \\underset {\\boldsymbol {U} _ {k _ {n}} ^ {T} \\boldsymbol {U} _ {k _ {n}} = \\boldsymbol {I}} {\\arg \\min } \\| \\boldsymbol {U} _ {k _ {n}} [ \\sqrt {\\mu^ {(t)}} \\boldsymbol {\\mathcal {B}} _ {(k _ {n})}, \\sqrt {\\eta^ {(t)}} \\boldsymbol {I} ] - [ \\sqrt {\\mu^ {(t)}} \\boldsymbol {\\mathcal {A}} _ {(k _ {n})}, \\sqrt {\\eta^ {(t)}} \\boldsymbol {U} _ {k _ {n}} ^ {(t)} ] \\| _ {F} ^ {2}.\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.241,
+ 0.768,
+ 0.789,
+ 0.809
+ ],
+ "angle": 0,
+ "content": "The optimal solution \\( \\pmb{U}_{k_n}^{(t + 1)} \\) can be gievn by \\( \\pmb{U}_{k_n}^{(t + 1)} = \\pmb{U}\\pmb{V}^T \\) from [29], where \\( \\pmb{U} \\) and \\( \\pmb{V} \\) can be obtained by SVD of \\( \\mu^{(t)}\\pmb{A}_{(k_n)}\\pmb{B}_{(k_n)}^T + \\eta^{(t)}\\pmb{U}_{k_n}^{(t)} \\):"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.244,
+ 0.809,
+ 0.51,
+ 0.828
+ ],
+ "angle": 0,
+ "content": "\\[\n\\boldsymbol {\\mu} ^ {(t)} \\boldsymbol {\\mathcal {A}} _ {(k _ {n})} \\boldsymbol {\\mathcal {B}} _ {(k _ {n})} ^ {T} + \\eta^ {(t)} \\boldsymbol {U} _ {k _ {n}} ^ {(t)} = \\boldsymbol {U} \\boldsymbol {\\Sigma} \\boldsymbol {V} ^ {T}.\n\\]"
+ }
+ ],
+ [
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.218,
+ 0.116,
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+ ],
+ "angle": 0,
+ "content": "8"
+ },
+ {
+ "type": "header",
+ "bbox": [
+ 0.272,
+ 0.115,
+ 0.37,
+ 0.129
+ ],
+ "angle": 0,
+ "content": "J. Zheng et al."
+ },
+ {
+ "type": "code_caption",
+ "bbox": [
+ 0.229,
+ 0.151,
+ 0.607,
+ 0.167
+ ],
+ "angle": 0,
+ "content": "Algorithm 1: APMM-based Iterative Solver to (8)"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.241,
+ 0.17,
+ 0.703,
+ 0.19
+ ],
+ "angle": 0,
+ "content": "Input: \\(\\Psi_{\\mathbb{I}}(\\mathcal{M}),\\{U_{k_n}^{(0)}\\}_{n = s + 1}^h,\\{\\hat{U}_{k_n}^{(0)}\\}_{n = 3}^h,\\pmb {\\varepsilon}^{(0)},\\pmb {\\mathcal{Y}}^{(0)},t = 0,\\mu^{(0)},\\eta^{(0)}\\)"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.294,
+ 0.19,
+ 0.432,
+ 0.203
+ ],
+ "angle": 0,
+ "content": "\\[\n\\rho_ {\\mu}, \\rho_ {\\eta} > 1, \\bar {\\mu}, \\text {a n d} \\bar {\\eta}.\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.241,
+ 0.202,
+ 0.551,
+ 0.219
+ ],
+ "angle": 0,
+ "content": "Output: \\(\\mathcal{Z}^{(t + 1)},\\{U_{k_n}^{(t + 1)}\\}_{n = s + 1}^h\\) , and \\(\\pmb{x}^{(t + 1)}\\)"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.241,
+ 0.219,
+ 0.434,
+ 0.231
+ ],
+ "angle": 0,
+ "content": "1. While not converge do"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.241,
+ 0.231,
+ 0.755,
+ 0.258
+ ],
+ "angle": 0,
+ "content": "2. \\(\\pmb{\\mathcal{Z}}^{(t + 1)} = \\arg \\min_{\\pmb{\\mathcal{Z}}}\\mathcal{L}(\\pmb {\\mathcal{Z}},\\{\\pmb{U}_{k_n}^{(t)}\\}_{n = s + 1}^h,\\pmb{\\mathcal{E}}^{(t)},\\pmb{\\mathcal{Y}}^{(t)},\\mu^{(t)}) + \\frac{\\eta^{(t)}}{2}\\| \\pmb{\\mathcal{Z}}^{(t)} - \\pmb {\\mathcal{Z}}\\| _F^2;\\)"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.241,
+ 0.257,
+ 0.69,
+ 0.278
+ ],
+ "angle": 0,
+ "content": "3. Calculate \\(\\pmb{U}_{k_n}^{(t + 1)} = \\arg \\min_{\\pmb{U}_{k_n}^T\\pmb{U}_{k_n} = \\pmb{I}}\\mathcal{L}(\\pmb{\\mathcal{Z}}^{(t + 1)},\\{\\pmb{U}_{k_n}^{(t + 1)}\\}_{n = s + 1}^{n - 1},\\)"
+ },
+ {
+ "type": "list",
+ "bbox": [
+ 0.241,
+ 0.219,
+ 0.755,
+ 0.278
+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.241,
+ 0.278,
+ 0.741,
+ 0.304
+ ],
+ "angle": 0,
+ "content": "\\[\n\\boldsymbol {U} _ {k _ {n}}, \\{\\boldsymbol {U} _ {k _ {n}} ^ {(t)} \\} _ {n = n + 1} ^ {h}, \\boldsymbol {\\mathcal {E}} ^ {(t)}, \\boldsymbol {\\mathcal {Y}} ^ {(t)}, \\mu^ {(t)}) + \\frac {\\eta^ {(t)}}{2} \\| \\boldsymbol {U} _ {k _ {n}} ^ {(t)} - \\boldsymbol {U} _ {k _ {n}} \\| _ {F} ^ {2}, (s + 1 \\leq n _ {0} \\leq h);\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.241,
+ 0.304,
+ 0.766,
+ 0.324
+ ],
+ "angle": 0,
+ "content": "4. Calculate \\(\\pmb{\\chi}^{(t + 1)}\\) by \\(\\pmb{\\chi}^{(t + 1)} = \\pmb{Z}^{(t + 1)}\\times_{k_h}\\pmb{U}_{k_h}^{(t + 1)T}\\times_{k_{h - 1}}\\dots \\times_{k_{s + 1}}\\pmb{U}_{k_{s + 1}}^{(t + 1)T}\\);"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.241,
+ 0.323,
+ 0.766,
+ 0.358
+ ],
+ "angle": 0,
+ "content": "\\[\n\\boldsymbol {\\mathcal {E}} ^ {(t + 1)} = \\arg \\min _ {\\boldsymbol {\\mathcal {E}} \\in \\mathbb {E}} \\mathcal {L} (\\boldsymbol {\\mathcal {Z}} ^ {(t + 1)}, \\{\\boldsymbol {U} _ {k _ {n}} ^ {(t + 1)} \\} _ {n = s + 1} ^ {h}, \\boldsymbol {\\mathcal {E}}, \\boldsymbol {\\mathcal {Y}} ^ {(t)}, \\mu^ {(t)}) + \\frac {\\eta^ {(t)}}{2} \\| \\boldsymbol {\\mathcal {E}} - \\boldsymbol {\\mathcal {E}} ^ {(t)} \\| _ {F} ^ {2};\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.241,
+ 0.357,
+ 0.725,
+ 0.372
+ ],
+ "angle": 0,
+ "content": "6. Calculate \\(\\pmb{y}^{(t + 1)}\\) by \\(\\pmb{y}^{(t + 1)} = \\mu^{(t)}(\\varPsi_{\\mathbb{I}}(\\mathcal{M}) - \\pmb{x}^{(t + 1)} - \\pmb{\\varepsilon}^{(t + 1)}) + \\pmb{y}^{(t)}\\)"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.241,
+ 0.371,
+ 0.763,
+ 0.387
+ ],
+ "angle": 0,
+ "content": "7. Calculate \\(\\mu^{(t + 1)} = \\min (\\bar{\\mu},\\rho_{\\mu}\\mu^{(t)})\\) and \\(\\eta^{(t + 1)} = \\min (\\bar{\\eta},\\rho_{\\eta}\\eta^{(t)})\\), respectively;"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.241,
+ 0.386,
+ 0.647,
+ 0.4
+ ],
+ "angle": 0,
+ "content": "8. Check the convergence condition: \\(\\| \\pmb{Z}^{(t + 1)} - \\pmb{Z}^{(t)}\\|_{\\infty} < \\varepsilon\\)"
+ },
+ {
+ "type": "list",
+ "bbox": [
+ 0.241,
+ 0.357,
+ 0.763,
+ 0.4
+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.251,
+ 0.4,
+ 0.405,
+ 0.414
+ ],
+ "angle": 0,
+ "content": "\\[\n\\| \\boldsymbol {\\mathcal {X}} ^ {(t + 1)} - \\boldsymbol {\\mathcal {X}} ^ {(t)} \\| _ {\\infty} < \\varepsilon ,\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.274,
+ 0.414,
+ 0.6,
+ 0.431
+ ],
+ "angle": 0,
+ "content": "\\[\n\\left\\| \\boldsymbol {U} _ {k _ {n}} ^ {(t + 1)} - \\boldsymbol {U} _ {k _ {n}} ^ {(t)} \\right\\| _ {\\infty} < \\varepsilon \\text {f o r} n = s + 1, s + 2, \\dots , h;\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.241,
+ 0.43,
+ 0.318,
+ 0.442
+ ],
+ "angle": 0,
+ "content": "9. \\(t = t + 1\\)"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.241,
+ 0.443,
+ 0.342,
+ 0.456
+ ],
+ "angle": 0,
+ "content": "10. end while"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.227,
+ 0.491,
+ 0.381,
+ 0.507
+ ],
+ "angle": 0,
+ "content": "Calculate \\(\\mathcal{E}^{(t + 1)}\\)"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.244,
+ 0.519,
+ 0.786,
+ 0.589
+ ],
+ "angle": 0,
+ "content": "\\[\n\\begin{array}{l} \\boldsymbol {\\mathcal {E}} ^ {(t + 1)} = \\underset {\\boldsymbol {\\mathcal {E}} \\in \\mathbb {E}} {\\arg \\min } \\frac {\\mu^ {(t)}}{2} \\| \\Psi_ {\\mathbb {I}} (\\boldsymbol {\\mathcal {M}}) - \\boldsymbol {\\mathcal {X}} ^ {(t + 1)} - \\boldsymbol {\\mathcal {E}} + \\frac {1}{\\mu^ {(t)}} \\boldsymbol {\\mathcal {Y}} ^ {(t)} \\| _ {F} ^ {2} + \\frac {\\eta^ {(t)}}{2} \\| \\boldsymbol {\\mathcal {E}} - \\boldsymbol {\\mathcal {E}} ^ {(t)} \\| _ {F} ^ {2} \\\\ = \\Psi_ {\\mathbb {I} ^ {c}} \\big (\\frac {1}{\\mu^ {(t)} + \\eta^ {(t)}} \\big (\\mu^ {(t)} (\\Psi_ {\\mathbb {I}} (\\boldsymbol {\\mathcal {M}}) - \\boldsymbol {\\mathcal {X}} ^ {(t + 1)} + \\frac {1}{\\mu^ {(t)}} \\boldsymbol {\\mathcal {Y}} ^ {(t)}) + \\eta^ {(t)} \\boldsymbol {\\mathcal {E}} ^ {(t)}) \\big). \\\\ \\end{array}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.216,
+ 0.628,
+ 0.788,
+ 0.743
+ ],
+ "angle": 0,
+ "content": "Computational Complexity The most time-consuming steps in the algorithm 1 are the computations of \\(\\mathcal{Z}\\), and \\(U_{k_n}\\). Since the computational complexity of n-mode product of \\(\\hat{\\mathcal{P}}\\in \\mathbb{R}^{I_1\\times I_2\\times \\dots \\times I_h}\\) and \\(U_{n}\\in \\mathbb{R}^{I_{n}\\times I_{n}}\\) is \\(\\mathcal{O}(I_nI_1I_2\\dots I_h)\\), the complexity of the computation of \\(\\mathcal{Z}\\) is \\(\\mathcal{O}(hI_{(1)}I_1I_2\\dots I_h)\\), where \\(I_{(1)} = \\max_k(I_k)\\). Besides, the complexity of the computation of \\(U_{k_n}\\) is \\(\\mathcal{O}((h - 1)I_{(1)}I_1I_2\\dots I_h + I_k^2 I_1I_2\\dots I_h)\\), therefore the overall computational complexity of each iteration of Algorithm 1 is \\(\\mathcal{O}\\Big((h - s)(h + I_{(1)} - 1)I_{(1)}I_1I_2\\dots I_h + hI_{(1)}I_1I_2\\dots I_h\\Big)\\)."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.214,
+ 0.761,
+ 0.789,
+ 0.824
+ ],
+ "angle": 0,
+ "content": "Convergence Analysis Although the optimization problem (8) is non-convex because of the constraints \\(\\pmb{U}_{k_n}^T\\pmb{U}_{k_n} = \\pmb{I}(n = s + 1,s + 2,\\dots ,h)\\) and the global optimality for (8) is hardly guaranteed, we can still prove some excellent convergence properties of the Algorithm 1, as stated in the following theorem."
+ }
+ ],
+ [
+ {
+ "type": "header",
+ "bbox": [
+ 0.377,
+ 0.115,
+ 0.733,
+ 0.129
+ ],
+ "angle": 0,
+ "content": "Handling The Non-Smooth Challenge in Tensor SVD"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.775,
+ 0.117,
+ 0.785,
+ 0.127
+ ],
+ "angle": 0,
+ "content": "9"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.214,
+ 0.145,
+ 0.788,
+ 0.211
+ ],
+ "angle": 0,
+ "content": "Theorem 1. For the sequence \\(\\{[\\pmb{\\mathcal{Z}}^{(t)},\\{\\pmb{U}_{k_n}^{(t)}\\}_{n = s + 1}^h,\\pmb{\\mathcal{E}}^{(t)},\\pmb{\\mathcal{V}}^{(t)},\\pmb{\\mu}^{(t)}]\\}\\) generated by the proposed algorithm 1, we have the following properties if \\(\\{\\pmb{\\mathcal{V}}^{(t)}\\}\\) is bounded, \\(\\sum_{t = 1}^{\\infty}(\\mu^{(t)})^{-2}\\mu^{(t + 1)} < + \\infty\\) and \\(\\lim_{n\\to \\infty}\\mu^{(n)}\\sum_{t = n}^{\\infty}(\\eta^{(t)})^{-1 / 2} = 0.\\)"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.217,
+ 0.224,
+ 0.784,
+ 0.268
+ ],
+ "angle": 0,
+ "content": "(i) \\(\\lim_{t\\to \\infty}\\Psi_{\\mathbb{I}}(\\boldsymbol {M}) - \\boldsymbol{x}^{(t)} - \\boldsymbol{\\varepsilon}^{(t)} = \\mathbf{0},\\) where \\(\\pmb {\\mathcal{X}}^{(t)} = \\pmb {\\mathcal{Z}}^{(t)}\\times_{k_h}(\\pmb{U}_{k_h}^{(t)})^T\\dots \\times_{k_{s + 1}}\\) \\((\\pmb{U}_{k_{s + 1}}^{(t)})^T\\)"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.212,
+ 0.27,
+ 0.565,
+ 0.289
+ ],
+ "angle": 0,
+ "content": "(ii) \\(\\{[\\pmb{\\mathcal{Z}}^{(t)},\\{\\pmb{U}_{k_n}^{(t)}\\}_{n = s + 1}^h ],\\pmb{\\mathcal{X}}^{(t)},\\pmb{\\mathcal{E}}^{(t)}]\\}\\) is bounded."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.209,
+ 0.29,
+ 0.786,
+ 0.323
+ ],
+ "angle": 0,
+ "content": "(iii) \\(\\sum_{t=1}^{\\infty} \\eta^{(t)} \\| [\\pmb{\\mathcal{Z}}^{(t)}, \\{\\pmb{U}_{k_n}^{(t)}\\}_{n=s+1}^h, \\pmb{\\mathcal{E}}^{(t)}] - [\\pmb{\\mathcal{Z}}^{(t+1)}, \\{\\pmb{U}_{k_n}^{(t+1)}\\}_{n=s+1}^h, \\pmb{\\mathcal{E}}^{(t+1)}] \\|_F^2\\) is convergent. Thus, we have"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.246,
+ 0.336,
+ 0.782,
+ 0.367
+ ],
+ "angle": 0,
+ "content": "\\[\n\\| [ \\boldsymbol {\\mathcal {Z}} ^ {(t)}, \\{\\boldsymbol {U} _ {k _ {n}} ^ {(t)} \\} _ {n = s + 1} ^ {h}, \\boldsymbol {\\mathcal {E}} ^ {(t)} ] - [ \\boldsymbol {\\mathcal {Z}} ^ {(t + 1)}, \\{\\boldsymbol {U} _ {k _ {n}} ^ {(t + 1)} \\} _ {n = s + 1} ^ {h}, \\boldsymbol {\\mathcal {E}} ^ {(t + 1)} ] \\| _ {F} ^ {2} \\leq \\mathcal {O} (\\frac {1}{\\eta^ {(t)}}).\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.212,
+ 0.38,
+ 0.449,
+ 0.404
+ ],
+ "angle": 0,
+ "content": "\\((iv)\\) \\(\\lim_{t\\to \\infty}\\| \\pmb{\\mathcal{V}}^{(t + 1)} - \\pmb{\\mathcal{V}}^{(t)}\\| _F = 0.\\)"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.214,
+ 0.405,
+ 0.785,
+ 0.444
+ ],
+ "angle": 0,
+ "content": "(v) Let \\([\\mathcal{Z}^*,\\{\\pmb{U}_{k_n}^*\\}_{n = s + 1},\\pmb {\\varepsilon}^*,\\pmb {\\mathcal{V}}^* ]\\) be any limit point of \\(\\{[\\pmb {\\mathcal{Z}}^{(t)},\\{\\pmb{U}_{k_n}^{(t)}\\}_{n = s + 1}^h,\\pmb {\\mathcal{E}}^{(t)},\\) \\(\\pmb {\\mathcal{V}}^{(t)}]\\}\\). Then, \\([\\pmb {\\mathcal{Z}}^*,\\{\\pmb{U}_{k_n}^*\\}_{n = s + 1}^h,\\pmb {\\mathcal{E}}^*,\\pmb {\\mathcal{V}}^* ]\\) is a KKT point to (8)."
+ },
+ {
+ "type": "list",
+ "bbox": [
+ 0.212,
+ 0.38,
+ 0.785,
+ 0.444
+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.214,
+ 0.457,
+ 0.788,
+ 0.552
+ ],
+ "angle": 0,
+ "content": "Please refer to the supplementary material of this paper for the proof of the Theorem 1. Theorem 1 shows that, if \\(\\{\\pmb{v}^{(t)}\\}\\) is bounded, the sequence \\([\\pmb{z}^{(t)},\\{\\pmb{U}_{k_n}^{(t)}\\}_{n = s + 1}^h,\\pmb{\\varepsilon}^{(t)}]\\) generated by the proposed algorithm 1 is Cauchy convergent, with a convergence rate of at least \\(\\mathcal{O}\\left(\\frac{1}{\\eta^{(t)}}\\right)\\). Moreover, any accumulation point of the sequence converges to the KKT point of (8)."
+ },
+ {
+ "type": "title",
+ "bbox": [
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+ 0.577,
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+ "angle": 0,
+ "content": "3.2 APMM-based Heuristic Method for Solving MOTC"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.214,
+ 0.608,
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+ 0.641
+ ],
+ "angle": 0,
+ "content": "To solve MOTC, we iteratively update \\( U_{k_n}^{(k_1,k_2)}(n = 1 + s,\\dots ,h,1\\leq k_1 < k_2\\leq h) \\) and \\( \\mathcal{X} \\) in (7) by solving the following optimization problems:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.387,
+ 0.652,
+ 0.787,
+ 0.686
+ ],
+ "angle": 0,
+ "content": "\\[\n\\hat {\\mathcal {U}} _ {(k _ {1}, k _ {2})} = \\underset {\\tilde {\\mathcal {U}} _ {(k _ {1}, k _ {2})}} {\\arg \\min } \\| \\hat {\\boldsymbol {X}} \\| _ {*}, \\tilde {\\mathcal {U}} _ {(k _ {1}, k _ {2})} ^ {[ k _ {1}, k _ {2} ]} \\tag {10}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ ],
+ "angle": 0,
+ "content": "and"
+ },
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+ "type": "equation",
+ "bbox": [
+ 0.289,
+ 0.711,
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+ ],
+ "angle": 0,
+ "content": "\\[\n\\hat {\\boldsymbol {\\mathcal {X}}} = \\underset {\\boldsymbol {\\mathcal {X}}} {\\arg \\min } \\left[ \\| \\boldsymbol {\\mathcal {X}} \\| _ {*} ^ {(k _ {1}, k _ {2})} \\right] _ {1 \\leq k _ {1} < k _ {2} \\leq h} s. t. \\Psi_ {\\mathbb {I}} (\\boldsymbol {\\mathcal {M}}) = \\Psi_ {\\mathbb {I}} (\\boldsymbol {\\mathcal {X}}), \\tag {11}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.214,
+ 0.75,
+ 0.788,
+ 0.841
+ ],
+ "angle": 0,
+ "content": "respectively. We use the proposed APMM for solving (10), and the non-dominated sorting genetic algorithm (NSGA-II) [5] presented in Algorithm 2 for solving (11). These two steps are used for the learning of the tensor nuclear norm functions and the low rank estimation, respectively. It is worth noting that both updating the \\(\\hat{\\mathcal{U}}_{(k_1,k_2)}\\) for each \\((k_{1},k_{2})\\) and evaluating the individuals in Algorithm 2 can be proceed in parallel, which can accelerate the whole optimization for MOTC."
+ }
+ ],
+ [
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.218,
+ 0.116,
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+ 0.127
+ ],
+ "angle": 0,
+ "content": "10"
+ },
+ {
+ "type": "header",
+ "bbox": [
+ 0.272,
+ 0.115,
+ 0.37,
+ 0.128
+ ],
+ "angle": 0,
+ "content": "J. Zheng et al."
+ },
+ {
+ "type": "code_caption",
+ "bbox": [
+ 0.229,
+ 0.151,
+ 0.652,
+ 0.167
+ ],
+ "angle": 0,
+ "content": "Algorithm 2: NSGA-II-based framework for solving (11)"
+ },
+ {
+ "type": "algorithm",
+ "bbox": [
+ 0.242,
+ 0.171,
+ 0.761,
+ 0.349
+ ],
+ "angle": 0,
+ "content": "Input: \\(\\Psi_{\\mathbb{I}}(\\mathcal{M}),\\{\\hat{\\mathcal{U}}_{(k_1,k_2)}\\}_{1\\leq k_1 < k_2\\leq h}\\) , IterMax, and \\(t = 0\\) \nOutput: \\(\\hat{\\pmb{x}}\\) \n1. Initialize: population \\(\\mathbb{P}^{(0)}\\) \n2. Evaluate \\(\\left[\\| \\pmb {x}\\|_{*,\\hat{\\pmb{u}}_{(k_1,k_2)}}^{[k_1,k_2]}\\right]_{1\\leq k_1 < k_2\\leq h}\\) for each \\(\\pmb {\\chi}\\in \\mathbb{P}^{(t)}\\) . \n3. Sorting individuals by their non-domination ranks and crowding distance, and obtaining the first front \\(\\mathbb{F}_1\\) . \n4. Selecting parents and applying crossover and mutation to create offspring; \n5. Obtaining the next generation by truncating the sorted individuals (including offspring population) and update the first front \\(\\mathbb{F}_1\\) . \n6. \\(t = t + 1\\) \n7. Repeat steps 2-6 until \\(t = \\mathrm{IterMax}\\) \n8. \\(\\hat{\\pmb{x}} =\\) average sum of the individuals that belong to \\(\\mathbb{F}_1\\)"
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.216,
+ 0.378,
+ 0.461,
+ 0.396
+ ],
+ "angle": 0,
+ "content": "4 Experimental Results"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.214,
+ 0.406,
+ 0.789,
+ 0.467
+ ],
+ "angle": 0,
+ "content": "In this section, we compared the TC-SL and MOTC| Notations | Descriptions | Notations | Descriptions |
| a, b, c, ... | calvars | a, b, c, ... | vectors |
| A, B, C, ... | matrices | A, B, C, ... | tensors |
| A, B, C, ... | sets | A, B, C, ... | operators |
| [1:n] | {1,2,...,n} | AT | transpose of A |
| I | identity tensor | 0 | null tensor |
| [At1,i2,...,ih] | (i1,i2,...,ih)-th element of A | ||A||0 | the number of non-zero elements of A |
| ||A||1 | ||A||1 = ∑i1,i2,...,ih ||A||i1,i2,...,ih| | ATk1,...,kh | rotation of A ∈ R1×...×Ih such that ATk1,...,kh ∈ Rl2×...×Ikh |
| [Atk1,...,kh], ...,ik3,...,ikh | slice along the (k1,k2)-th mode | A(k) (or [Aj(n)]) | Mode-n Unfolding of A |
| ×n | Mode-n product | Ac | complementary set of A |
+
+# 2.1 Tensor Completion Problem
+
+Considering tensor data $\mathcal{M} \in \mathbb{R}^{I_1 \times I_2 \times \dots \times I_h}$ may with missing elements, [17, 20] propose the following t-SVD-based tensor completion model:
+
+$$
+\min _ {\boldsymbol {x}} \operatorname {r a n k} _ {[ k _ {1}, k _ {2} ]} (\boldsymbol {x}) \quad s. t. \Psi_ {\mathbb {I}} (\boldsymbol {\mathcal {M}}) = \Psi_ {\mathbb {I}} (\boldsymbol {x}), \tag {1}
+$$
+
+where $\varPsi_{\mathbb{I}}$ is a linear project operator on the support set $\mathbb{I}$ such that
+
+$$
+[ \varPsi_{\mathbb{I}}(\mathcal{M})]_{i_{1},i_{2},\dots ,i_{h}}=\left\{\begin{array}{ll}[\mathcal{M}]_{i_{1},i_{2},\cdots,i_{h}},& \mbox{if } (i_{1},i_{2},\cdots,i_{h})\in\mathbb{I};\\ 0,&\mbox{if }(i_{1},i_{2},\cdots,i_{h})\notin\mathbb{I},\end{array}\right.
+$$
+
+and $i_k \in [1: I_k]$ for $k = 1, 2, \dots, h$ . The tensor rank function in (1) operates on the assumption that an $h$ -order tensor data $\mathcal{M} \in \mathbb{R}^{I_1 \times I_2 \times \dots \times I_h}$ in the real world can be decomposed as
+
+$$
+\boldsymbol {\mathcal {M}} = \boldsymbol {\mathcal {Z}} \times_ {k _ {3}} \hat {\boldsymbol {U}} _ {k _ {3}} ^ {T} \dots \times_ {k _ {h}} \hat {\boldsymbol {U}} _ {k _ {h}} ^ {T}, \tag {2}
+$$
+
+where $\{k_3, k_4, \dots, k_h\} \subset [1:h]$ , $\{\hat{U}_{k_n}\}_{n=3}^h$ are a set of given invertible transforms, and
+
+$$
+\begin{array}{l} \operatorname {r a n k} _ {[ k _ {1}, k _ {2} ]} (\boldsymbol {\mathcal {M}}) = \max _ {i _ {k _ {n}} \in [ 1: I _ {k _ {n}} ] \text {f o r} n = 3, 4, \dots , h} \operatorname {r a n k} \left(\left[ \boldsymbol {\mathcal {Z}} ^ {T _ {k _ {1}}, \dots , k _ {h}} \right] _ {:,: i _ {k _ {3}}, \dots , i _ {k _ {h}}}\right) \\ < < \min \left(I _ {k _ {1}}, I _ {k _ {2}}\right) \tag {3} \\ \end{array}
+$$
+
+for certain $(k_{1},k_{2})$ satisfying $1\leq k_{1} < k_{2}\leq h$ . If taken $\{\hat{U}_{k_n}\}_{n = 3}^h$ as Discrete Fourier Matrices, $\mathrm{rank}_{[k_1,k_2]}(\mathcal{M})$ is referred as to tensor tubal rank of $\mathcal{M}$ [16].
+
+# 2.2 The Proposed Tensor Completion with A Learnable Tensor Rank Function
+
+From the definition of $\mathrm{rank}_{[k_1,k_2]}(\mathcal{M})$ , we can see that it aims to examine the slice-wise tensor rank along the $(k_{1},k_{2})$ -th mode of transformed data $\pmb{\mathcal{Z}}$ . It allows (1) to investigate the low-rankness of different features in tensor data corresponding to different frequencies separately and jointly.
+
+However, when $\mathcal{M}$ exhibits harsh changes caused by disordered tensor slices along a certain mode $k_{n}$ , all information in $\mathcal{M}$ tends to collapse into the high-frequency slices in $\mathcal{Z}$ . To handle such situation, we introduce a learnable permutation matrix $P\in \mathbb{R}^{I_{k_n}\times I_{k_n}}$ to (2):
+
+$$
+\pmb {\mathcal {M}} \times_ {k _ {n}} \pmb {P} = \pmb {\mathcal {Z}} \times_ {k _ {3}} \hat {\pmb {U}} _ {k _ {3}} ^ {T} \dots \times_ {k _ {n}} (\hat {\pmb {U}} _ {k _ {n}} \pmb {P}) ^ {T} \times_ {k _ {n + 1}} \dots \times_ {k _ {h}} \hat {\pmb {U}} _ {k _ {h}} ^ {T},
+$$
+
+and therefore we can get the following model:
+
+$$
+\begin{array}{l} \min _ {\boldsymbol {X}, \boldsymbol {P} _ {k _ {n}}} \operatorname {r a n k} _ {[ k _ {1}, k _ {2} ]} (\boldsymbol {X} \times_ {k _ {n}} \boldsymbol {P} _ {k _ {n}}) \\ s. t. \Psi_ {\mathbb {I}} (\boldsymbol {\mathcal {M}}) = \Psi_ {\mathbb {I}} (\boldsymbol {\mathcal {X}}), \\ \sum_ {i = 1} ^ {I _ {k _ {n}}} \left[ \boldsymbol {P} _ {k _ {n}} \right] _ {i, j} = 1, \quad \sum_ {j = 1} ^ {I _ {k _ {n}}} \left[ \boldsymbol {P} _ {k _ {n}} \right] _ {i, j} = 1 \text {f o r} \left[ \boldsymbol {P} _ {k _ {n}} \right] _ {i, j} \in \{0, 1 \}. \tag {4} \\ \end{array}
+$$
+
+This incorporation is aimed at addressing the slice permutation property in tensor completion methods, thus facilitating a more effective exploration of the low-rank property in $\mathcal{M}$ .
+
+Unfortunately, solving (4) is challenging due to the constraints of $\sum_{i=1}^{I_{k_n}}[P_{k_n}]_{i,j} = 1$ and $\sum_{j=1}^{I_{k_n}}[P_{k_n}]_{i,j} = 1$ for $[P_{k_n}]_{i,j} \in \{0,1\}$ . Therefore, we opt to use a set of learnable unitary matrices $\{\pmb{U}_{k_n}\}_{n=s+1}^h$ instead, proposing the following tensor completion model for given $\{k_{s+1}, k_{s+2}, \dots, k_h\}$ :
+
+$$
+\begin{array}{l} \min _ {\boldsymbol {X}, \boldsymbol {U} _ {k _ {n}} (n = s + 1, \dots , h)} \operatorname {r a n k} _ {[ k _ {1}, k _ {2} ]} (\boldsymbol {\mathcal {X}} \times_ {k _ {s + 1}} \boldsymbol {U} _ {k _ {s + 1}} \dots \times_ {k _ {h}} \boldsymbol {U} _ {k _ {h}}) \\ s. t. \Psi_ {\mathbb {I}} (\boldsymbol {\mathcal {M}}) = \Psi_ {\mathbb {I}} (\boldsymbol {\mathcal {X}}), \boldsymbol {U} _ {k _ {n}} ^ {T} \boldsymbol {U} _ {k _ {n}} = \boldsymbol {I} (n = s + 1, \dots , h), \tag {5} \\ \end{array}
+$$
+
+where $\mathcal{X}$ is a low-rank estimation of the true tensor data $\mathcal{M}$ . Besides simplifying the optimization process of (4), the introduction of the learnable unitary matrices enables the model to extract features for better studying the low-rankness in tensor data and to handle other scenarios where the initially provided transforms $\{\hat{U}_{k_n}\}_{n=3}^h$ are inadequate, such as videos with irregularly changing content and image sequences with different scenes. Hence, we introduce our learnable tensor rank as
+
+$$
+\mathrm {r a n k} _ {[ k _ {1}, k _ {2} ], \tilde {\mathcal {U}}} (\pmb {\mathscr {X}}) = \mathrm {r a n k} _ {[ k _ {1}, k _ {2} ]} (\tilde {\mathcal {U}} (\pmb {\mathscr {X}})),
+$$
+
+where $\tilde{\mathcal{U}} (\pmb {x}) = \pmb {x}\times_{k_{s + 1}}\pmb{U}_{k_{s + 1}}\dots \times_{k_h}\pmb{U}_{k_h}.$
+
+# 2.3 Approximation to The Proposed Tensor Completion by Using A Learnable Tensor Nuclear Norm (TC-SL)
+
+Since the function $\mathrm{rank}_{[k_1,k_2],\hat{\mathcal{U}}}(\cdot)$ is discrete, it often leads to the NP-hard problem. From the result given in [16,17], we know that, when $h = 3$ and $\hat{U}_3$ is orthogonal, $\| \cdot \|_{*}^{[1,2]}$ is the tightest convex envelope of $\mathrm{rank}_{[1,2]}(\cdot)$ on the set $\{\mathcal{A}||\mathcal{A}||_2^{[1,2]}\leq 1\}$ , where the definitions of $\| \cdot \| _2^{[k_1,k_2]}$ and $\| \cdot \|_{*}^{[k_{1},k_{2}]}$ are given in Definition 1. This conclusion can be easily extended to our case, and we obtain the Property 1.
+
+Definition 1. For an $h$ -order tensor $\mathcal{A} \in \mathbb{R}^{I_1 \times I_2 \times \dots \times I_h}$ , $\| \mathcal{A} \|_*^{[k_1, k_2]}$ and $\| \mathcal{A} \|_2^{[k_1, k_2]}$ are defined as
+
+$$
+\| \boldsymbol {\mathcal {A}} \| _ {*} ^ {[ k _ {1}, k _ {2} ]} = \sum_ {i _ {k _ {3}}, i _ {k _ {4}}, \dots , i _ {k _ {h}}} \| [ \boldsymbol {\mathcal {A}} ^ {T _ {k _ {1}}, \dots , k _ {h}} \times_ {k _ {3}} \hat {\boldsymbol {U}} _ {k _ {3}} \cdot \cdot \cdot \times_ {k _ {h}} \hat {\boldsymbol {U}} _ {k _ {h}} ] _ {:;,: i _ {k _ {3}}, \dots , i _ {k _ {h}}} \| _ {*}
+$$
+
+and
+
+$$
+\| \pmb {\mathcal {A}} \| _ {2} ^ {[ k _ {1}, k _ {2} ]} = \max _ {i _ {k _ {3}}, i _ {k _ {4}}, \dots , i _ {k _ {h}}} \| [ \pmb {\mathcal {A}} ^ {T _ {k _ {1}}, \dots , k _ {h}} \times_ {k _ {3}} \hat {\pmb {U}} _ {k _ {3}} \cdot \cdot \cdot \times_ {k _ {h}} \hat {\pmb {U}} _ {k _ {h}} ] _ {:,: i _ {k _ {3}}, \dots , i _ {k _ {h}}} \| _ {2},
+$$
+
+respectively.
+
+Property 1. For an h-order tensor $\mathcal{A} \in \mathbb{R}^{I_1 \times I_2 \times \dots \times I_h}$ , let us define $\| \mathcal{A} \|_{*,\tilde{\mathcal{U}}}^{[k_1,k_2]}$ and $\| \mathcal{A} \|_{2,\tilde{\mathcal{U}}}^{[k_1,k_2]}$ are defined as $\| \mathcal{A} \|_{*,\tilde{\mathcal{U}}}^{[k_1,k_2]} = \| \tilde{\mathcal{U}} (\mathcal{A}) \|_{*}^{[k_1,k_2]}$ and $\| \mathcal{A} \|_{2,\tilde{\mathcal{U}}}^{[k_1,k_2]} = \| \tilde{\mathcal{U}} (\mathcal{A}) \|_{2}^{[k_1,k_2]}$ , respectively. $\| \cdot \|_{*,\tilde{\mathcal{U}}}^{[k_1,k_2]}$ is the dual norm of tensor $\| \cdot \|_{2,\tilde{\mathcal{U}}}^{[k_1,k_2]}$ norm, and $\| \cdot \|_{*,\tilde{\mathcal{U}}}^{[k_1,k_2]}$ is the tightest convex envelope of $\mathrm{rank}_{[k_1,k_2],\tilde{\mathcal{U}}}(\cdot)$ on the set $\{\mathcal{A} \| \mathcal{A} \|_{2,\tilde{\mathcal{U}}}^{[k_1,k_2]} \leq 1\}$ .
+
+Therefore, we derive an approximation to the proposed model (6) by utilizing a learnable tensor nuclear norm based on the prior assumption of slice-wise low-rankness in the transformed data (TC-SL):
+
+$$
+\min _ {\boldsymbol {X}, \tilde {\mathcal {U}}} \| \boldsymbol {X} \| _ {*}, \tilde {\mathcal {U}} ^ {[ k _ {1}, k _ {2} ]} \quad s. t. \Psi_ {\mathbb {I}} (\boldsymbol {\mathcal {M}}) = \Psi_ {\mathbb {I}} (\boldsymbol {\mathcal {X}}). \tag {6}
+$$
+
+Let $\hat{\mathcal{U}}$ be the invertible transform operator learned by (6). The exactly recovery of TC-SL with given $\hat{\mathcal{U}}$ is guaranteed from the current studies in the exactly recovery of the t-SVD-based tensor completion [16, 17].
+
+# 2.4 The Proposed Multi-Objective Tensor Completion to Learn The Cross-Dimensional Low-Rankness
+
+From (6), we observe that the definition of TC-SL depends on the choice of $k_{1}$ and $k_{2}$ , and it considers different kinds of low-rankness in the tensor data by adjusting $(k_{1}, k_{2})$ . However, considering only one mode may result in the loss of correlation information across the remaining modes. To address this issue, we give the following multi-objective model with learnable tensor nuclear norms for tensor completion (MOTC):
+
+$$
+\min _ {\boldsymbol {x}, \tilde {\mathcal {U}} _ {\left(k _ {1}, k _ {2}\right)}} \left[ \| \boldsymbol {\mathcal {X}} \| _ {*}, \tilde {\mathcal {U}} _ {\left(k _ {1}, k _ {2}\right)} ^ {[ k _ {1}, k _ {2} ]} \right] _ {1 \leq k _ {1} < k _ {2} \leq h} \quad s. t. \Psi_ {\mathbb {I}} (\boldsymbol {\mathcal {M}}) = \Psi_ {\mathbb {I}} (\boldsymbol {\mathcal {X}}), \tag {7}
+$$
+
+where $\tilde{\mathcal{U}}_{(k_1,k_2)}(\boldsymbol {x}) = \boldsymbol {x}\times_{k_{s + 1}}\boldsymbol{U}_{k_{s + 1}}^{(k_1,k_2)}\dots \times_{k_h}\boldsymbol{U}_{k_h}^{(k_1,k_2)}$ and $(\boldsymbol{U}_{k_n}^{(k_1,k_2)})^T\boldsymbol{U}_{k_n}^{(k_1,k_2)} = \boldsymbol{I}$ for $n = 1 + s,\dots ,h$ and $1\leq k_{1} < k_{2}\leq h$ . In MOTC, the multiple objective functions $\| \pmb {x}\|_{*,\tilde{\mathcal{U}}_{(k_1,k_2)}}^{[k_1,k_2]}$ $(1\leq k_{1} < k_{2}\leq h)$ are utilized to examine the low-rankness of tensor data from its various dimensions.
+
+# 3 Optimization Algorithm
+
+In this section, we provide a detailed discussion for the optimizing (6) and (7).
+
+# 3.1 Alternating Proximal Multiplier Method (APMM) for TC-SL
+
+To solve the problem (6), we introduce auxiliary variables $\pmb{\mathcal{E}}\in \mathbb{E} = \{\pmb {\mathcal{E}}|\Psi_{\mathbb{I}}(\pmb {\mathcal{E}}) = \mathbf{0}\}$ and $\pmb{\Xi}$ such that $\pmb {x} = \pmb {z}\times_{k_{s + 1}}\pmb{U}_{k_{s + 1}}^T\dots \times_{k_h}\pmb{U}_{k_h}^T$ . Therefore, we turn to solve the following equivalence problem:
+
+$$
+\min _ {\boldsymbol {Z}, \boldsymbol {U} _ {k _ {n}} ^ {T} \boldsymbol {U} _ {k _ {n}} = \boldsymbol {I} (n = s + 1, \dots , h)} \| \boldsymbol {Z} \| _ {*} ^ {[ k _ {1}, k _ {2} ]} \quad \text {s . t .} \Psi_ {\mathbb {I}} (\boldsymbol {\mathcal {M}}) = \boldsymbol {\mathcal {Z}} \times_ {k _ {s + 1}} \boldsymbol {U} _ {k _ {s + 1}} ^ {T} \dots \times_ {k _ {h}} \boldsymbol {U} _ {k _ {h}} ^ {T} + \boldsymbol {\mathcal {E}}. \tag {8}
+$$
+
+The Augmented Lagrangian function of (8) is formulated as
+
+$$
+\begin{array}{l} \mathcal {L} (\boldsymbol {\mathcal {Z}}, \{\boldsymbol {U} _ {k _ {n}} \} _ {n = s + 1} ^ {h}, \boldsymbol {\mathcal {E}}, \boldsymbol {\mathcal {Y}}, \mu) = \| \boldsymbol {\mathcal {Z}} \| _ {*} ^ {[ k _ {1}, k _ {2} ]} + \left\langle \Psi_ {\mathbb {I}} (\boldsymbol {\mathcal {M}}) - \boldsymbol {\mathcal {Z}} \times_ {k _ {s + 1}} \boldsymbol {U} _ {k _ {s + 1}} ^ {T} \dots \times_ {k _ {h}} \boldsymbol {U} _ {k _ {h}} ^ {T} \right. \\ - \mathcal {E}, \mathcal {Y} \rangle + \frac {\mu}{2} \| \Psi_ {\mathbb {I}} (\mathcal {M}) - \mathcal {Z} \times_ {k _ {s + 1}} \boldsymbol {U} _ {k _ {s + 1}} ^ {T} \dots \times_ {k _ {h}} \boldsymbol {U} _ {k _ {h}} ^ {T} - \mathcal {E} \| _ {F} ^ {2}, \tag {9} \\ \end{array}
+$$
+
+where $\mathcal{V}$ is Lagrange multiplier, and $\mu$ is a positive scalar. We solve (8) iteratively by combining the proximal algorithm with the Alternating Direction Method of Multipliers (APMM) that is given in the Algorithm 1. We detail the solutions for solving $\mathcal{Z}^{(t + 1)}, U_{k_n}^{(t + 1)}$ , and $\mathcal{E}^{(t + 1)}$ as follows.
+
+Calculate $\mathcal{Z}^{(t + 1)}$
+
+$$
+\begin{array}{l} \boldsymbol {\mathcal {Z}} ^ {(t + 1)} = \underset {\boldsymbol {\mathcal {Z}}} {\arg \min } \frac {1}{2} \| \frac {\mu^ {(t)} \hat {\boldsymbol {\mathcal {P}}} \times_ {k _ {s + 1}} \boldsymbol {U} _ {k _ {s + 1}} ^ {(t)} \cdots \times_ {k _ {h}} \boldsymbol {U} _ {k _ {h}} ^ {(t)} + \eta^ {(t)} \boldsymbol {\mathcal {Z}} ^ {(t)}}{\mu^ {(t)} + \eta^ {(t)}} - \boldsymbol {\mathcal {Z}} \| _ {F} ^ {2} \\ + \frac {1}{\mu^ {(t)} + \eta^ {(t)}} \| \mathcal {Z} \| _ {*} ^ {[ k _ {1}, k _ {2} ]}, \\ \end{array}
+$$
+
+where $\hat{\mathcal{P}} = \Psi_{\mathbb{I}}(\mathcal{M}) - \pmb{\mathcal{E}}^{(t)} + \frac{1}{\mu^{(t)}}\pmb{\mathcal{Y}}^{(t)}$ . It can be solved by the tensor singular value thresholding operation with parameter $\frac{1}{\mu^{(t)} + \eta^{(t)}}$ [15].
+
+Calculate $U_{k_n}^{(t + 1)}$ : Let $\mathcal{A} = \mathcal{Z}^{(t + 1)} \times_{k_h} U_{k_h}^{(t)T} \times_{k_h - 1} \dots \times_{k_n + 1} U_{k_n + 1}^{(t)T}$ and $\mathcal{B} = \hat{\mathcal{P}} \times_{s + 1} U_{k_{s + 1}}^{(t + 1)} \dots \times_{k_n - 1} U_{k_n - 1}^{(t + 1)}$ , and we have
+
+$$
+\boldsymbol {U} _ {k _ {n}} ^ {(t + 1)} = \underset {\boldsymbol {U} _ {k _ {n}} ^ {T} \boldsymbol {U} _ {k _ {n}} = \boldsymbol {I}} {\arg \min } \| \boldsymbol {U} _ {k _ {n}} [ \sqrt {\mu^ {(t)}} \boldsymbol {\mathcal {B}} _ {(k _ {n})}, \sqrt {\eta^ {(t)}} \boldsymbol {I} ] - [ \sqrt {\mu^ {(t)}} \boldsymbol {\mathcal {A}} _ {(k _ {n})}, \sqrt {\eta^ {(t)}} \boldsymbol {U} _ {k _ {n}} ^ {(t)} ] \| _ {F} ^ {2}.
+$$
+
+The optimal solution $\pmb{U}_{k_n}^{(t + 1)}$ can be gievn by $\pmb{U}_{k_n}^{(t + 1)} = \pmb{U}\pmb{V}^T$ from [29], where $\pmb{U}$ and $\pmb{V}$ can be obtained by SVD of $\mu^{(t)}\pmb{A}_{(k_n)}\pmb{B}_{(k_n)}^T + \eta^{(t)}\pmb{U}_{k_n}^{(t)}$ :
+
+$$
+\boldsymbol {\mu} ^ {(t)} \boldsymbol {\mathcal {A}} _ {(k _ {n})} \boldsymbol {\mathcal {B}} _ {(k _ {n})} ^ {T} + \eta^ {(t)} \boldsymbol {U} _ {k _ {n}} ^ {(t)} = \boldsymbol {U} \boldsymbol {\Sigma} \boldsymbol {V} ^ {T}.
+$$
+
+Algorithm 1: APMM-based Iterative Solver to (8)
+
+Input: $\Psi_{\mathbb{I}}(\mathcal{M}),\{U_{k_n}^{(0)}\}_{n = s + 1}^h,\{\hat{U}_{k_n}^{(0)}\}_{n = 3}^h,\pmb {\varepsilon}^{(0)},\pmb {\mathcal{Y}}^{(0)},t = 0,\mu^{(0)},\eta^{(0)}$
+
+$$
+\rho_ {\mu}, \rho_ {\eta} > 1, \bar {\mu}, \text {a n d} \bar {\eta}.
+$$
+
+Output: $\mathcal{Z}^{(t + 1)},\{U_{k_n}^{(t + 1)}\}_{n = s + 1}^h$ , and $\pmb{x}^{(t + 1)}$
+
+1. While not converge do
+2. $\pmb{\mathcal{Z}}^{(t + 1)} = \arg \min_{\pmb{\mathcal{Z}}}\mathcal{L}(\pmb {\mathcal{Z}},\{\pmb{U}_{k_n}^{(t)}\}_{n = s + 1}^h,\pmb{\mathcal{E}}^{(t)},\pmb{\mathcal{Y}}^{(t)},\mu^{(t)}) + \frac{\eta^{(t)}}{2}\| \pmb{\mathcal{Z}}^{(t)} - \pmb {\mathcal{Z}}\| _F^2;$
+3. Calculate $\pmb{U}_{k_n}^{(t + 1)} = \arg \min_{\pmb{U}_{k_n}^T\pmb{U}_{k_n} = \pmb{I}}\mathcal{L}(\pmb{\mathcal{Z}}^{(t + 1)},\{\pmb{U}_{k_n}^{(t + 1)}\}_{n = s + 1}^{n - 1},$
+
+$$
+\boldsymbol {U} _ {k _ {n}}, \{\boldsymbol {U} _ {k _ {n}} ^ {(t)} \} _ {n = n + 1} ^ {h}, \boldsymbol {\mathcal {E}} ^ {(t)}, \boldsymbol {\mathcal {Y}} ^ {(t)}, \mu^ {(t)}) + \frac {\eta^ {(t)}}{2} \| \boldsymbol {U} _ {k _ {n}} ^ {(t)} - \boldsymbol {U} _ {k _ {n}} \| _ {F} ^ {2}, (s + 1 \leq n _ {0} \leq h);
+$$
+
+4. Calculate $\pmb{\chi}^{(t + 1)}$ by $\pmb{\chi}^{(t + 1)} = \pmb{Z}^{(t + 1)}\times_{k_h}\pmb{U}_{k_h}^{(t + 1)T}\times_{k_{h - 1}}\dots \times_{k_{s + 1}}\pmb{U}_{k_{s + 1}}^{(t + 1)T}$ ;
+
+$$
+\boldsymbol {\mathcal {E}} ^ {(t + 1)} = \arg \min _ {\boldsymbol {\mathcal {E}} \in \mathbb {E}} \mathcal {L} (\boldsymbol {\mathcal {Z}} ^ {(t + 1)}, \{\boldsymbol {U} _ {k _ {n}} ^ {(t + 1)} \} _ {n = s + 1} ^ {h}, \boldsymbol {\mathcal {E}}, \boldsymbol {\mathcal {Y}} ^ {(t)}, \mu^ {(t)}) + \frac {\eta^ {(t)}}{2} \| \boldsymbol {\mathcal {E}} - \boldsymbol {\mathcal {E}} ^ {(t)} \| _ {F} ^ {2};
+$$
+
+6. Calculate $\pmb{y}^{(t + 1)}$ by $\pmb{y}^{(t + 1)} = \mu^{(t)}(\varPsi_{\mathbb{I}}(\mathcal{M}) - \pmb{x}^{(t + 1)} - \pmb{\varepsilon}^{(t + 1)}) + \pmb{y}^{(t)}$
+7. Calculate $\mu^{(t + 1)} = \min (\bar{\mu},\rho_{\mu}\mu^{(t)})$ and $\eta^{(t + 1)} = \min (\bar{\eta},\rho_{\eta}\eta^{(t)})$ , respectively;
+8. Check the convergence condition: $\| \pmb{Z}^{(t + 1)} - \pmb{Z}^{(t)}\|_{\infty} < \varepsilon$
+
+$$
+\| \boldsymbol {\mathcal {X}} ^ {(t + 1)} - \boldsymbol {\mathcal {X}} ^ {(t)} \| _ {\infty} < \varepsilon ,
+$$
+
+$$
+\left\| \boldsymbol {U} _ {k _ {n}} ^ {(t + 1)} - \boldsymbol {U} _ {k _ {n}} ^ {(t)} \right\| _ {\infty} < \varepsilon \text {f o r} n = s + 1, s + 2, \dots , h;
+$$
+
+9. $t = t + 1$
+
+10. end while
+
+Calculate $\mathcal{E}^{(t + 1)}$
+
+$$
+\begin{array}{l} \boldsymbol {\mathcal {E}} ^ {(t + 1)} = \underset {\boldsymbol {\mathcal {E}} \in \mathbb {E}} {\arg \min } \frac {\mu^ {(t)}}{2} \| \Psi_ {\mathbb {I}} (\boldsymbol {\mathcal {M}}) - \boldsymbol {\mathcal {X}} ^ {(t + 1)} - \boldsymbol {\mathcal {E}} + \frac {1}{\mu^ {(t)}} \boldsymbol {\mathcal {Y}} ^ {(t)} \| _ {F} ^ {2} + \frac {\eta^ {(t)}}{2} \| \boldsymbol {\mathcal {E}} - \boldsymbol {\mathcal {E}} ^ {(t)} \| _ {F} ^ {2} \\ = \Psi_ {\mathbb {I} ^ {c}} \big (\frac {1}{\mu^ {(t)} + \eta^ {(t)}} \big (\mu^ {(t)} (\Psi_ {\mathbb {I}} (\boldsymbol {\mathcal {M}}) - \boldsymbol {\mathcal {X}} ^ {(t + 1)} + \frac {1}{\mu^ {(t)}} \boldsymbol {\mathcal {Y}} ^ {(t)}) + \eta^ {(t)} \boldsymbol {\mathcal {E}} ^ {(t)}) \big). \\ \end{array}
+$$
+
+Computational Complexity The most time-consuming steps in the algorithm 1 are the computations of $\mathcal{Z}$ , and $U_{k_n}$ . Since the computational complexity of n-mode product of $\hat{\mathcal{P}}\in \mathbb{R}^{I_1\times I_2\times \dots \times I_h}$ and $U_{n}\in \mathbb{R}^{I_{n}\times I_{n}}$ is $\mathcal{O}(I_nI_1I_2\dots I_h)$ , the complexity of the computation of $\mathcal{Z}$ is $\mathcal{O}(hI_{(1)}I_1I_2\dots I_h)$ , where $I_{(1)} = \max_k(I_k)$ . Besides, the complexity of the computation of $U_{k_n}$ is $\mathcal{O}((h - 1)I_{(1)}I_1I_2\dots I_h + I_k^2 I_1I_2\dots I_h)$ , therefore the overall computational complexity of each iteration of Algorithm 1 is $\mathcal{O}\Big((h - s)(h + I_{(1)} - 1)I_{(1)}I_1I_2\dots I_h + hI_{(1)}I_1I_2\dots I_h\Big)$ .
+
+Convergence Analysis Although the optimization problem (8) is non-convex because of the constraints $\pmb{U}_{k_n}^T\pmb{U}_{k_n} = \pmb{I}(n = s + 1,s + 2,\dots ,h)$ and the global optimality for (8) is hardly guaranteed, we can still prove some excellent convergence properties of the Algorithm 1, as stated in the following theorem.
+
+Theorem 1. For the sequence $\{[\pmb{\mathcal{Z}}^{(t)},\{\pmb{U}_{k_n}^{(t)}\}_{n = s + 1}^h,\pmb{\mathcal{E}}^{(t)},\pmb{\mathcal{V}}^{(t)},\pmb{\mu}^{(t)}]\}$ generated by the proposed algorithm 1, we have the following properties if $\{\pmb{\mathcal{V}}^{(t)}\}$ is bounded, $\sum_{t = 1}^{\infty}(\mu^{(t)})^{-2}\mu^{(t + 1)} < + \infty$ and $\lim_{n\to \infty}\mu^{(n)}\sum_{t = n}^{\infty}(\eta^{(t)})^{-1 / 2} = 0.$
+
+(i) $\lim_{t\to \infty}\Psi_{\mathbb{I}}(\boldsymbol {M}) - \boldsymbol{x}^{(t)} - \boldsymbol{\varepsilon}^{(t)} = \mathbf{0},$ where $\pmb {\mathcal{X}}^{(t)} = \pmb {\mathcal{Z}}^{(t)}\times_{k_h}(\pmb{U}_{k_h}^{(t)})^T\dots \times_{k_{s + 1}}$ $(\pmb{U}_{k_{s + 1}}^{(t)})^T$
+
+(ii) $\{[\pmb{\mathcal{Z}}^{(t)},\{\pmb{U}_{k_n}^{(t)}\}_{n = s + 1}^h ],\pmb{\mathcal{X}}^{(t)},\pmb{\mathcal{E}}^{(t)}]\}$ is bounded.
+
+(iii) $\sum_{t=1}^{\infty} \eta^{(t)} \| [\pmb{\mathcal{Z}}^{(t)}, \{\pmb{U}_{k_n}^{(t)}\}_{n=s+1}^h, \pmb{\mathcal{E}}^{(t)}] - [\pmb{\mathcal{Z}}^{(t+1)}, \{\pmb{U}_{k_n}^{(t+1)}\}_{n=s+1}^h, \pmb{\mathcal{E}}^{(t+1)}] \|_F^2$ is convergent. Thus, we have
+
+$$
+\| [ \boldsymbol {\mathcal {Z}} ^ {(t)}, \{\boldsymbol {U} _ {k _ {n}} ^ {(t)} \} _ {n = s + 1} ^ {h}, \boldsymbol {\mathcal {E}} ^ {(t)} ] - [ \boldsymbol {\mathcal {Z}} ^ {(t + 1)}, \{\boldsymbol {U} _ {k _ {n}} ^ {(t + 1)} \} _ {n = s + 1} ^ {h}, \boldsymbol {\mathcal {E}} ^ {(t + 1)} ] \| _ {F} ^ {2} \leq \mathcal {O} (\frac {1}{\eta^ {(t)}}).
+$$
+
+$(iv)$ $\lim_{t\to \infty}\| \pmb{\mathcal{V}}^{(t + 1)} - \pmb{\mathcal{V}}^{(t)}\| _F = 0.$
+(v) Let $[\mathcal{Z}^*,\{\pmb{U}_{k_n}^*\}_{n = s + 1},\pmb {\varepsilon}^*,\pmb {\mathcal{V}}^* ]$ be any limit point of $\{[\pmb {\mathcal{Z}}^{(t)},\{\pmb{U}_{k_n}^{(t)}\}_{n = s + 1}^h,\pmb {\mathcal{E}}^{(t)},$ $\pmb {\mathcal{V}}^{(t)}]\}$ . Then, $[\pmb {\mathcal{Z}}^*,\{\pmb{U}_{k_n}^*\}_{n = s + 1}^h,\pmb {\mathcal{E}}^*,\pmb {\mathcal{V}}^* ]$ is a KKT point to (8).
+
+Please refer to the supplementary material of this paper for the proof of the Theorem 1. Theorem 1 shows that, if $\{\pmb{v}^{(t)}\}$ is bounded, the sequence $[\pmb{z}^{(t)},\{\pmb{U}_{k_n}^{(t)}\}_{n = s + 1}^h,\pmb{\varepsilon}^{(t)}]$ generated by the proposed algorithm 1 is Cauchy convergent, with a convergence rate of at least $\mathcal{O}\left(\frac{1}{\eta^{(t)}}\right)$ . Moreover, any accumulation point of the sequence converges to the KKT point of (8).
+
+# 3.2 APMM-based Heuristic Method for Solving MOTC
+
+To solve MOTC, we iteratively update $U_{k_n}^{(k_1,k_2)}(n = 1 + s,\dots ,h,1\leq k_1 < k_2\leq h)$ and $\mathcal{X}$ in (7) by solving the following optimization problems:
+
+$$
+\hat {\mathcal {U}} _ {(k _ {1}, k _ {2})} = \underset {\tilde {\mathcal {U}} _ {(k _ {1}, k _ {2})}} {\arg \min } \| \hat {\boldsymbol {X}} \| _ {*}, \tilde {\mathcal {U}} _ {(k _ {1}, k _ {2})} ^ {[ k _ {1}, k _ {2} ]} \tag {10}
+$$
+
+and
+
+$$
+\hat {\boldsymbol {\mathcal {X}}} = \underset {\boldsymbol {\mathcal {X}}} {\arg \min } \left[ \| \boldsymbol {\mathcal {X}} \| _ {*} ^ {(k _ {1}, k _ {2})} \right] _ {1 \leq k _ {1} < k _ {2} \leq h} s. t. \Psi_ {\mathbb {I}} (\boldsymbol {\mathcal {M}}) = \Psi_ {\mathbb {I}} (\boldsymbol {\mathcal {X}}), \tag {11}
+$$
+
+respectively. We use the proposed APMM for solving (10), and the non-dominated sorting genetic algorithm (NSGA-II) [5] presented in Algorithm 2 for solving (11). These two steps are used for the learning of the tensor nuclear norm functions and the low rank estimation, respectively. It is worth noting that both updating the $\hat{\mathcal{U}}_{(k_1,k_2)}$ for each $(k_{1},k_{2})$ and evaluating the individuals in Algorithm 2 can be proceed in parallel, which can accelerate the whole optimization for MOTC.
+
+Algorithm 2: NSGA-II-based framework for solving (11)
+Input: $\Psi_{\mathbb{I}}(\mathcal{M}),\{\hat{\mathcal{U}}_{(k_1,k_2)}\}_{1\leq k_1 < k_2\leq h}$ , IterMax, and $t = 0$
+Output: $\hat{\pmb{x}}$
+1. Initialize: population $\mathbb{P}^{(0)}$
+2. Evaluate $\left[\| \pmb {x}\|_{*,\hat{\pmb{u}}_{(k_1,k_2)}}^{[k_1,k_2]}\right]_{1\leq k_1 < k_2\leq h}$ for each $\pmb {\chi}\in \mathbb{P}^{(t)}$ .
+3. Sorting individuals by their non-domination ranks and crowding distance, and obtaining the first front $\mathbb{F}_1$ .
+4. Selecting parents and applying crossover and mutation to create offspring;
+5. Obtaining the next generation by truncating the sorted individuals (including offspring population) and update the first front $\mathbb{F}_1$ .
+6. $t = t + 1$
+7. Repeat steps 2-6 until $t = \mathrm{IterMax}$
+8. $\hat{\pmb{x}} =$ average sum of the individuals that belong to $\mathbb{F}_1$
+
+# 4 Experimental Results
+
+In this section, we compared the TC-SL and MOTC| Notations | Descriptions | Notations | Descriptions |
| a, b, c, ... | calvars | a, b, c, ... | vectors |
| A, B, C, ... | matrices | A, B, C, ... | tensors |
| A, B, C, ... | sets | A, B, C, ... | operators |
| [1:n] | {1,2,...,n} | AT | transpose of A |
| I | identity tensor | 0 | null tensor |
| [At1,i2,...,ih] | (i1,i2,...,ih)-th element of A | ||A||0 | the number of non-zero elements of A |
| ||A||1 | ||A||1 = ∑i1,i2,...,ih ||A||i1,i2,...,ih| | ATk1,...,kh | rotation of A ∈ R1×...×Ih such that ATk1,...,kh ∈ Rl2×...×Ikh |
| [Atk1,...,kh], ...,ik3,...,ikh | slice along the (k1,k2)-th mode | A(k) (or [Aj(n)]) | Mode-n Unfolding of A |
| ×n | Mode-n product | Ac | complementary set of A |
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+ "content": "\\pmb{\\mathcal{E}}\\in \\mathbb{E} = \\{\\pmb {\\mathcal{E}}|\\Psi_{\\mathbb{I}}(\\pmb {\\mathcal{E}}) = \\mathbf{0}\\}"
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+ "content": "\\pmb {x} = \\pmb {z}\\times_{k_{s + 1}}\\pmb{U}_{k_{s + 1}}^T\\dots \\times_{k_h}\\pmb{U}_{k_h}^T"
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+ "type": "text",
+ "content": " . Therefore, we turn to solve the following equivalence problem:"
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+ "content": "\\min _ {\\boldsymbol {Z}, \\boldsymbol {U} _ {k _ {n}} ^ {T} \\boldsymbol {U} _ {k _ {n}} = \\boldsymbol {I} (n = s + 1, \\dots , h)} \\| \\boldsymbol {Z} \\| _ {*} ^ {[ k _ {1}, k _ {2} ]} \\quad \\text {s . t .} \\Psi_ {\\mathbb {I}} (\\boldsymbol {\\mathcal {M}}) = \\boldsymbol {\\mathcal {Z}} \\times_ {k _ {s + 1}} \\boldsymbol {U} _ {k _ {s + 1}} ^ {T} \\dots \\times_ {k _ {h}} \\boldsymbol {U} _ {k _ {h}} ^ {T} + \\boldsymbol {\\mathcal {E}}. \\tag {8}",
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+ "content": "The Augmented Lagrangian function of (8) is formulated as"
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+ "content": "\\begin{array}{l} \\mathcal {L} (\\boldsymbol {\\mathcal {Z}}, \\{\\boldsymbol {U} _ {k _ {n}} \\} _ {n = s + 1} ^ {h}, \\boldsymbol {\\mathcal {E}}, \\boldsymbol {\\mathcal {Y}}, \\mu) = \\| \\boldsymbol {\\mathcal {Z}} \\| _ {*} ^ {[ k _ {1}, k _ {2} ]} + \\left\\langle \\Psi_ {\\mathbb {I}} (\\boldsymbol {\\mathcal {M}}) - \\boldsymbol {\\mathcal {Z}} \\times_ {k _ {s + 1}} \\boldsymbol {U} _ {k _ {s + 1}} ^ {T} \\dots \\times_ {k _ {h}} \\boldsymbol {U} _ {k _ {h}} ^ {T} \\right. \\\\ - \\mathcal {E}, \\mathcal {Y} \\rangle + \\frac {\\mu}{2} \\| \\Psi_ {\\mathbb {I}} (\\mathcal {M}) - \\mathcal {Z} \\times_ {k _ {s + 1}} \\boldsymbol {U} _ {k _ {s + 1}} ^ {T} \\dots \\times_ {k _ {h}} \\boldsymbol {U} _ {k _ {h}} ^ {T} - \\mathcal {E} \\| _ {F} ^ {2}, \\tag {9} \\\\ \\end{array}",
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+ "content": " is Lagrange multiplier, and "
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+ "type": "text",
+ "content": " is a positive scalar. We solve (8) iteratively by combining the proximal algorithm with the Alternating Direction Method of Multipliers (APMM) that is given in the Algorithm 1. We detail the solutions for solving "
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+ "type": "inline_equation",
+ "content": "\\mathcal{Z}^{(t + 1)}, U_{k_n}^{(t + 1)}"
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+ "type": "text",
+ "content": " as follows."
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+ "content": "Calculate "
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+ "type": "inline_equation",
+ "content": "\\mathcal{Z}^{(t + 1)}"
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+ "content": "\\begin{array}{l} \\boldsymbol {\\mathcal {Z}} ^ {(t + 1)} = \\underset {\\boldsymbol {\\mathcal {Z}}} {\\arg \\min } \\frac {1}{2} \\| \\frac {\\mu^ {(t)} \\hat {\\boldsymbol {\\mathcal {P}}} \\times_ {k _ {s + 1}} \\boldsymbol {U} _ {k _ {s + 1}} ^ {(t)} \\cdots \\times_ {k _ {h}} \\boldsymbol {U} _ {k _ {h}} ^ {(t)} + \\eta^ {(t)} \\boldsymbol {\\mathcal {Z}} ^ {(t)}}{\\mu^ {(t)} + \\eta^ {(t)}} - \\boldsymbol {\\mathcal {Z}} \\| _ {F} ^ {2} \\\\ + \\frac {1}{\\mu^ {(t)} + \\eta^ {(t)}} \\| \\mathcal {Z} \\| _ {*} ^ {[ k _ {1}, k _ {2} ]}, \\\\ \\end{array}",
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+ "content": "\\hat{\\mathcal{P}} = \\Psi_{\\mathbb{I}}(\\mathcal{M}) - \\pmb{\\mathcal{E}}^{(t)} + \\frac{1}{\\mu^{(t)}}\\pmb{\\mathcal{Y}}^{(t)}"
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+ "type": "text",
+ "content": ". It can be solved by the tensor singular value thresholding operation with parameter "
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+ "type": "inline_equation",
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+ "type": "text",
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+ "content": "\\mathcal{A} = \\mathcal{Z}^{(t + 1)} \\times_{k_h} U_{k_h}^{(t)T} \\times_{k_h - 1} \\dots \\times_{k_n + 1} U_{k_n + 1}^{(t)T}"
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+ "content": "\\mathcal{B} = \\hat{\\mathcal{P}} \\times_{s + 1} U_{k_{s + 1}}^{(t + 1)} \\dots \\times_{k_n - 1} U_{k_n - 1}^{(t + 1)}"
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+ "type": "text",
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+ "content": "\\boldsymbol {U} _ {k _ {n}} ^ {(t + 1)} = \\underset {\\boldsymbol {U} _ {k _ {n}} ^ {T} \\boldsymbol {U} _ {k _ {n}} = \\boldsymbol {I}} {\\arg \\min } \\| \\boldsymbol {U} _ {k _ {n}} [ \\sqrt {\\mu^ {(t)}} \\boldsymbol {\\mathcal {B}} _ {(k _ {n})}, \\sqrt {\\eta^ {(t)}} \\boldsymbol {I} ] - [ \\sqrt {\\mu^ {(t)}} \\boldsymbol {\\mathcal {A}} _ {(k _ {n})}, \\sqrt {\\eta^ {(t)}} \\boldsymbol {U} _ {k _ {n}} ^ {(t)} ] \\| _ {F} ^ {2}.",
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+ "content": " can be gievn by "
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+ "content": "\\pmb{U}_{k_n}^{(t + 1)} = \\pmb{U}\\pmb{V}^T"
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+ "content": " from [29], where "
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+ "content": " can be obtained by SVD of "
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+ "content": "\\mu^{(t)}\\pmb{A}_{(k_n)}\\pmb{B}_{(k_n)}^T + \\eta^{(t)}\\pmb{U}_{k_n}^{(t)}"
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+ "content": "\\boldsymbol {\\mu} ^ {(t)} \\boldsymbol {\\mathcal {A}} _ {(k _ {n})} \\boldsymbol {\\mathcal {B}} _ {(k _ {n})} ^ {T} + \\eta^ {(t)} \\boldsymbol {U} _ {k _ {n}} ^ {(t)} = \\boldsymbol {U} \\boldsymbol {\\Sigma} \\boldsymbol {V} ^ {T}.",
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+ "type": "inline_equation",
+ "content": "\\Psi_{\\mathbb{I}}(\\mathcal{M}),\\{U_{k_n}^{(0)}\\}_{n = s + 1}^h,\\{\\hat{U}_{k_n}^{(0)}\\}_{n = 3}^h,\\pmb {\\varepsilon}^{(0)},\\pmb {\\mathcal{Y}}^{(0)},t = 0,\\mu^{(0)},\\eta^{(0)}"
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+ "content": "\\rho_ {\\mu}, \\rho_ {\\eta} > 1, \\bar {\\mu}, \\text {a n d} \\bar {\\eta}.",
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+ "type": "inline_equation",
+ "content": "\\mathcal{Z}^{(t + 1)},\\{U_{k_n}^{(t + 1)}\\}_{n = s + 1}^h"
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+ "content": " , and "
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+ "type": "inline_equation",
+ "content": "\\pmb{x}^{(t + 1)}"
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+ "type": "text",
+ "content": "1. While not converge do"
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+ "content": "2. "
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+ "type": "inline_equation",
+ "content": "\\pmb{\\mathcal{Z}}^{(t + 1)} = \\arg \\min_{\\pmb{\\mathcal{Z}}}\\mathcal{L}(\\pmb {\\mathcal{Z}},\\{\\pmb{U}_{k_n}^{(t)}\\}_{n = s + 1}^h,\\pmb{\\mathcal{E}}^{(t)},\\pmb{\\mathcal{Y}}^{(t)},\\mu^{(t)}) + \\frac{\\eta^{(t)}}{2}\\| \\pmb{\\mathcal{Z}}^{(t)} - \\pmb {\\mathcal{Z}}\\| _F^2;"
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+ "content": "3. Calculate "
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+ "type": "inline_equation",
+ "content": "\\pmb{U}_{k_n}^{(t + 1)} = \\arg \\min_{\\pmb{U}_{k_n}^T\\pmb{U}_{k_n} = \\pmb{I}}\\mathcal{L}(\\pmb{\\mathcal{Z}}^{(t + 1)},\\{\\pmb{U}_{k_n}^{(t + 1)}\\}_{n = s + 1}^{n - 1},"
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+ "content": "\\boldsymbol {U} _ {k _ {n}}, \\{\\boldsymbol {U} _ {k _ {n}} ^ {(t)} \\} _ {n = n + 1} ^ {h}, \\boldsymbol {\\mathcal {E}} ^ {(t)}, \\boldsymbol {\\mathcal {Y}} ^ {(t)}, \\mu^ {(t)}) + \\frac {\\eta^ {(t)}}{2} \\| \\boldsymbol {U} _ {k _ {n}} ^ {(t)} - \\boldsymbol {U} _ {k _ {n}} \\| _ {F} ^ {2}, (s + 1 \\leq n _ {0} \\leq h);",
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+ "content": "\\pmb{\\chi}^{(t + 1)} = \\pmb{Z}^{(t + 1)}\\times_{k_h}\\pmb{U}_{k_h}^{(t + 1)T}\\times_{k_{h - 1}}\\dots \\times_{k_{s + 1}}\\pmb{U}_{k_{s + 1}}^{(t + 1)T}"
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+ "content": "\\boldsymbol {\\mathcal {E}} ^ {(t + 1)} = \\arg \\min _ {\\boldsymbol {\\mathcal {E}} \\in \\mathbb {E}} \\mathcal {L} (\\boldsymbol {\\mathcal {Z}} ^ {(t + 1)}, \\{\\boldsymbol {U} _ {k _ {n}} ^ {(t + 1)} \\} _ {n = s + 1} ^ {h}, \\boldsymbol {\\mathcal {E}}, \\boldsymbol {\\mathcal {Y}} ^ {(t)}, \\mu^ {(t)}) + \\frac {\\eta^ {(t)}}{2} \\| \\boldsymbol {\\mathcal {E}} - \\boldsymbol {\\mathcal {E}} ^ {(t)} \\| _ {F} ^ {2};",
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+ "type": "inline_equation",
+ "content": "\\mu^{(t + 1)} = \\min (\\bar{\\mu},\\rho_{\\mu}\\mu^{(t)})"
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+ "type": "text",
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+ "content": "\\eta^{(t + 1)} = \\min (\\bar{\\eta},\\rho_{\\eta}\\eta^{(t)})"
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+ "type": "text",
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+ "content": "Calculate "
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+ "type": "inline_equation",
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+ }
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+ "type": "text",
+ "content": "Computational Complexity The most time-consuming steps in the algorithm 1 are the computations of "
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+ "type": "inline_equation",
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+ "type": "inline_equation",
+ "content": "U_{k_n}"
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+ "type": "text",
+ "content": ". Since the computational complexity of n-mode product of "
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+ "type": "inline_equation",
+ "content": "\\hat{\\mathcal{P}}\\in \\mathbb{R}^{I_1\\times I_2\\times \\dots \\times I_h}"
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+ "content": " and "
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+ "content": "U_{n}\\in \\mathbb{R}^{I_{n}\\times I_{n}}"
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+ "type": "text",
+ "content": " is "
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+ "type": "inline_equation",
+ "content": "\\mathcal{O}(I_nI_1I_2\\dots I_h)"
+ },
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+ "type": "text",
+ "content": ", the complexity of the computation of "
+ },
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+ "type": "inline_equation",
+ "content": "\\mathcal{Z}"
+ },
+ {
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+ "type": "text",
+ "content": " is "
+ },
+ {
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+ ],
+ "type": "inline_equation",
+ "content": "\\mathcal{O}(hI_{(1)}I_1I_2\\dots I_h)"
+ },
+ {
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+ "type": "text",
+ "content": ", where "
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+ "type": "inline_equation",
+ "content": "I_{(1)} = \\max_k(I_k)"
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+ "type": "text",
+ "content": ". Besides, the complexity of the computation of "
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+ "type": "inline_equation",
+ "content": "U_{k_n}"
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+ "type": "text",
+ "content": " is "
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+ "type": "inline_equation",
+ "content": "\\mathcal{O}((h - 1)I_{(1)}I_1I_2\\dots I_h + I_k^2 I_1I_2\\dots I_h)"
+ },
+ {
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+ ],
+ "type": "text",
+ "content": ", therefore the overall computational complexity of each iteration of Algorithm 1 is "
+ },
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+ "type": "inline_equation",
+ "content": "\\mathcal{O}\\Big((h - s)(h + I_{(1)} - 1)I_{(1)}I_1I_2\\dots I_h + hI_{(1)}I_1I_2\\dots I_h\\Big)"
+ },
+ {
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+ "type": "text",
+ "content": "."
+ }
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+ }
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+ {
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+ "type": "text",
+ "content": "Convergence Analysis Although the optimization problem (8) is non-convex because of the constraints "
+ },
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+ "type": "inline_equation",
+ "content": "\\pmb{U}_{k_n}^T\\pmb{U}_{k_n} = \\pmb{I}(n = s + 1,s + 2,\\dots ,h)"
+ },
+ {
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+ "type": "text",
+ "content": " and the global optimality for (8) is hardly guaranteed, we can still prove some excellent convergence properties of the Algorithm 1, as stated in the following theorem."
+ }
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+ }
+ ],
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+ }
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+ "type": "text",
+ "content": "J. Zheng et al."
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+ "type": "text",
+ "content": "Theorem 1. For the sequence "
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+ "type": "inline_equation",
+ "content": "\\{[\\pmb{\\mathcal{Z}}^{(t)},\\{\\pmb{U}_{k_n}^{(t)}\\}_{n = s + 1}^h,\\pmb{\\mathcal{E}}^{(t)},\\pmb{\\mathcal{V}}^{(t)},\\pmb{\\mu}^{(t)}]\\}"
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+ "type": "text",
+ "content": " generated by the proposed algorithm 1, we have the following properties if "
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+ "type": "inline_equation",
+ "content": "\\{\\pmb{\\mathcal{V}}^{(t)}\\}"
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+ "type": "text",
+ "content": " is bounded, "
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+ "type": "inline_equation",
+ "content": "\\sum_{t = 1}^{\\infty}(\\mu^{(t)})^{-2}\\mu^{(t + 1)} < + \\infty"
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+ "type": "text",
+ "content": " and "
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+ "type": "inline_equation",
+ "content": "\\lim_{n\\to \\infty}\\mu^{(n)}\\sum_{t = n}^{\\infty}(\\eta^{(t)})^{-1 / 2} = 0."
+ }
+ ]
+ }
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+ "content": "(i) "
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+ ],
+ "type": "inline_equation",
+ "content": "\\lim_{t\\to \\infty}\\Psi_{\\mathbb{I}}(\\boldsymbol {M}) - \\boldsymbol{x}^{(t)} - \\boldsymbol{\\varepsilon}^{(t)} = \\mathbf{0},"
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+ ],
+ "type": "text",
+ "content": " where "
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+ ],
+ "type": "inline_equation",
+ "content": "\\pmb {\\mathcal{X}}^{(t)} = \\pmb {\\mathcal{Z}}^{(t)}\\times_{k_h}(\\pmb{U}_{k_h}^{(t)})^T\\dots \\times_{k_{s + 1}}"
+ },
+ {
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+ ],
+ "type": "inline_equation",
+ "content": "(\\pmb{U}_{k_{s + 1}}^{(t)})^T"
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+ ]
+ }
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+ "index": 3
+ },
+ {
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+ "content": "\\hat {\\boldsymbol {\\mathcal {X}}} = \\underset {\\boldsymbol {\\mathcal {X}}} {\\arg \\min } \\left[ \\| \\boldsymbol {\\mathcal {X}} \\| _ {*} ^ {(k _ {1}, k _ {2})} \\right] _ {1 \\leq k _ {1} < k _ {2} \\leq h} s. t. \\Psi_ {\\mathbb {I}} (\\boldsymbol {\\mathcal {M}}) = \\Psi_ {\\mathbb {I}} (\\boldsymbol {\\mathcal {X}}), \\tag {11}",
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+ "content": "respectively. We use the proposed APMM for solving (10), and the non-dominated sorting genetic algorithm (NSGA-II) [5] presented in Algorithm 2 for solving (11). These two steps are used for the learning of the tensor nuclear norm functions and the low rank estimation, respectively. It is worth noting that both updating the "
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+ "content": " and evaluating the individuals in Algorithm 2 can be proceed in parallel, which can accelerate the whole optimization for MOTC."
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+ "content": "In this section, we compared the TC-SL and MOTC | Caption | Paragraph |
| Kosmos-2 | A man is walking an elephant on a leash in a park. | a large elephant (0.308) is walking through a park (0.516), with people walking behind it (0.454). The elephant (0.308) is walking on a wet sidewalk (0.858), ... In the background, there are trees (0.679) and a building, adding to the park atmosphere. \nUnmatched: building, atmosphere |
| CIDEr | SPICE | InfoMetric |
| 1.18 | 0.20 | 9.24 |
| CLIPScore | BLIPScore | HiFi-Score |
| 0.87 | 0.44 | 1.40 |
| InstructBUP | An elephant walking across a wet pavement in front of people. | In the image, a large elephant (0.308) is standing in the middle of a park (0.585), surrounded by people (0.523) ... The elephant (0.308) appears to be interacting with the people, possibly as part of an exhibit or show. A few benches (0.570) ... \nUnmatched: interacting, exhibit |
| CIDEr | SPICE | InfoMetric |
| 1.34 | 0.14 | 12.39 |
| CLIPScore | BLIPScore | HiFi-Score |
| 0.90 | 0.47 | 1.47 |
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+ "1. Aditya, S., Yang, Y., Baral, C., Aloimonos, Y., Fermüller, C.: Image understanding using vision and reasoning through scene description graph. Computer Vision and Image Understanding 173, 33-45 (2018)",
+ "2. Alayrac, J.B., Donahue, J., Luc, P., Miech, A., Barr, I., Hasson, Y., Lenc, K., Mensch, A., Millican, K., Reynolds, M., Ring, R., Rutherford, E., Cabi, S., Han, T., Gong, Z., Samangooei, S., Monteiro, M., Menick, J., Borgeaud, S., Brock, A., Nematzadeh, A., Sharifzadeh, S., Binkowski, M., Barreira, R., Vinyals, O., Zisserman, A., Simonyan, K.: Flamingo: a visual language model for few-shot learning. In: Proceedings of the 36th International Conference on Neural Information Processing Systems (NeurIPS). pp. 23716-23736 (2022)",
+ "3. Anderson, P., Fernando, B., Johnson, M., Gould, S.: SPICE: Semantic propositional image caption evaluation. In: Proceedings of the European conference on computer vision (ECCV). pp. 382-398 (2016)",
+ "4. Banerjee, S., Lavie, A.: METEOR: An automatic metric for mt evaluation with improved correlation with human judgments. In: Proceedings of the ACL Workshop on Intrinsic and Extrinsic Evaluation Measures for Machine Translation and/or Summarization. pp. 65-72 (2005)",
+ "5. Chan, D., Petryk, S., Gonzalez, J., Darrell, T., Canny, J.: CLAIR: Evaluating image captions with large language models. In: Proceedings of the 2023 Conference on Empirical Methods in Natural Language Processing (EMNLP). pp. 13638-13646 (2023)",
+ "6. Chen, Y.C., Li, L., Yu, L., El Kholy, A., Ahmed, F., Gan, Z., Cheng, Y., Liu, J.: UNITER: Universal image-text representation learning. In: Proceedings of the European conference on computer vision (ECCV). pp. 104-120 (2020)",
+ "7. Cornia, M., Baraldi, L., Cucchiara, R.: Show, control and tell: A framework for generating controllable and grounded captions. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). pp. 8307-8316 (2019)",
+ "8. Cui, Y., Yang, G., Veit, A., Huang, X., Belongie, S.: Learning to evaluate image captioning. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). pp. 5804-5812 (2018)",
+ "9. Dai, W., Li, J., Li, D., Tiong, A., Zhao, J., Wang, W., Li, B., Fung, P., Hoi, S.: InstructBLIP: Towards general-purpose vision-language models with instruction tuning. In: Proceedings of the 37th International Conference on Neural Information Processing Systems (NeurIPS). pp. 49250-49267 (2023)",
+ "0. Deng, C., Ding, N., Tan, M., Wu, Q.: Length-controllable image captioning. In: Proceedings of the European conference on computer vision (ECCV). pp. 712-729 (2020)",
+ "1. Deshpande, A., Aneja, J., Wang, L., Schwing, A.G., Forsyth, D.: Fast, diverse and accurate image captioning guided by part-of-speech. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). pp. 10695-10704 (2019)"
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+ "12. Devlin, J., Chang, M.W., Lee, K., Toutanova, K.: BERT: Pre-training of deep bidirectional transformers for language understanding. In: Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics (NAACL). pp. 4171-4186 (2019)",
+ "13. Hessel, J., Holtzman, A., Forbes, M., Le Bras, R., Choi, Y.: CLIPScore: A reference-free evaluation metric for image captioning. In: Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing (EMNLP). pp. 7514-7528 (2021)",
+ "14. Hodosh, M., Young, P., Hockenmaier, J.: Framing image description as a ranking task: Data, models and evaluation metrics. Journal of Artificial Intelligence Research 47, 853-899 (2013)",
+ "15. Hu, A., Chen, S., Zhang, L., Jin, Q.: InfoMetIC: An informative metric for reference-free image caption evaluation. In: Proceedings of the 61st Annual Meeting of the Association for Computational Linguistics (ACL). pp. 3171-3185 (2023)",
+ "16. Jain, J., Li, J., Chiu, M.T., Hassani, A., Orlov, N., Shi, H.: OneFormer: One transformer to rule universal image segmentation. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). pp. 2989-2998 (2023)",
+ "17. Jiang, M., Hu, J., Huang, Q., Zhang, L., Diesner, J., Gao, J.: REO-relevance, extraness, omission: A fine-grained evaluation for image captioning. In: Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP). pp. 1475-1480 (2019)",
+ "18. Jiang, M., Huang, Q., Zhang, L., Wang, X., Zhang, P., Gan, Z., Diesner, J., Gao, J.: TIGEr: Text-to-image grounding for image caption evaluation. In: Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP). pp. 2141-2152 (2019)",
+ "19. Johnson, J., Karpathy, A., Fei-Fei, L.: DenseCap: Fully convolutional localization networks for dense captioning. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). pp. 4565-4574 (2016)",
+ "20. Kasai, J., Sakaguchi, K., Dunagan, L., Morrison, J., Bras, R.L., Choi, Y., Smith, N.A.: Transparent human evaluation for image captioning. In: Proceedings of the 2022 Conference of the North American Chapter of the Association for Computational Linguistics (NAACL). pp. 3464-3478 (2022)",
+ "21. Kim, D.J., Choi, J., Oh, T.H., Kweon, I.S.: Dense relational captioning: Triple-stream networks for relationship-based captioning. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). pp. 6271-6280 (2019)",
+ "22. Kirillov, A., Mintun, E., Ravi, N., Mao, H., Rolland, C., Gustafson, L., Xiao, T., Whitehead, S., Berg, A.C., Lo, W.Y., et al.: Segment anything. arXiv preprint arXiv:2304.02643 (2023)",
+ "23. Krause, J., Johnson, J., Krishna, R., Fei-Fei, L.: A hierarchical approach for generating descriptive image paragraphs. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). pp. 317-325 (2017)",
+ "24. Krishna, R., Zhu, Y., Groth, O., Johnson, J., Hata, K., Kravitz, J., Chen, S., Kalantidis, Y., Li, L.J., Shamma, D.A., Bernstein, M.S., Fei-Fei, L.: Visual Genome: Connecting language and vision using crowdsourced dense image annotations. International Journal of Computer Vision 123, 32-73 (2017)"
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+ "type": "list",
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+ "25. Kusner, M., Sun, Y., Kolkin, N., Weinberger, K.: From word embeddings to document distances. In: Proceedings of the 32nd International Conference on Machine Learning (ICML). pp. 957-966 (2015)",
+ "26. Lee, H., Yoon, S., Dernoncourt, F., Bui, T., Jung, K.: UMIC: An unreferenced metric for image captioning via contrastive learning. In: Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing (ACL-IJCNLP). pp. 220-226 (2021)",
+ "27. Lee, K.H., Chen, X., Hua, G., Hu, H., He, X.: Stacked cross attention for imagetext matching. In: Proceedings of the European conference on computer vision (ECCV). pp. 201-216 (2018)",
+ "28. Li, J., Li, D., Savarese, S., Hoi, S.: BLIP-2: bootstrapping language-image pretraining with frozen image encoders and large language models. In: Proceedings of the 40th International Conference on Machine Learning (ICML). pp. 19730-19742 (2023)",
+ "29. Li, L.H., Zhang, P., Zhang, H., Yang, J., Li, C., Zhong, Y., Wang, L., Yuan, L., Zhang, L., Hwang, J.N., et al.: Grounded language-image pre-training. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). pp. 10965-10975 (2022)",
+ "30. Lin, C.Y.: ROUGE: A package for automatic evaluation of summaries. In: Text Summarization Branches Out. pp. 74-81 (2004)",
+ "31. Liu, H., Li, C., Wu, Q., Lee, Y.J.: Visual instruction tuning. In: Proceedings of the 37th International Conference on Neural Information Processing Systems (NeurIPS). pp. 34892-34916 (2023)",
+ "32. Lu, Y., Yang, X., Li, X., Wang, X.E., Wang, W.Y.: LLMScore: Unveiling the power of large language models in text-to-image synthesis evaluation. In: Proceedings of the 37th International Conference on Neural Information Processing Systems (NeurIPS). pp. 23075-23093 (2023)",
+ "33. Madhyastha, P., Wang, J., Specia, L.: VIFIDEL: Evaluating the visual fidelity of image descriptions. In: Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics (ACL). pp. 6539-6550 (2019)",
+ "34. Navon, D.: Forest before trees: The precedence of global features in visual perception. Cognitive Psychology 9(3), 353-383 (1977)",
+ "35. Papineni, K., Roukos, S., Ward, T., Zhu, W.J.: BLEU: A method for automatic evaluation of machine translation. In: Proceedings of the 40th Annual Meeting of the Association for Computational Linguistics (ACL). pp. 311-318 (2002)",
+ "36. Peng, Z., Wang, W., Dong, L., Hao, Y., Huang, S., Ma, S., Ye, Q., Wei, F.: Grounding multimodal large language models to the world. In: Proceedings of the 12th International Conference on Learning Representations (ICLR) (2024)",
+ "37. Pont-Tuset, J., Uijlings, J., Changpinyo, S., Soricut, R., Ferrari, V.: Connecting vision and language with localized narratives. In: Proceedings of the European conference on computer vision (ECCV). pp. 647-664 (2020)",
+ "38. Radford, A., Kim, J.W., Hallacy, C., Ramesh, A., Goh, G., Agarwal, S., Sastry, G., Askell, A., Mishkin, P., Clark, J., et al.: Learning transferable visual models from natural language supervision. In: Proceedings of the 38th International Conference on Machine Learning. pp. 8748-8763 (2021)",
+ "39. Sarto, S., Barraco, M., Cornia, M., Baraldi, L., Cucchiara, R.: Positive-augmented contrastive learning for image and video captioning evaluation. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). pp. 6914-6924 (2023)"
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+ "40. Sharif, N., Bennamoun, M., White, L.R., Shah, S.A.A.: Learning-based composite metrics for improved caption evaluation. In: Proceedings of ACL 2018, Student Research Workshop. pp. 14-20 (2018)",
+ "41. Vedantam, R., Lawrence Zitnick, C., Parikh, D.: CIDEr: Consensus-based image description evaluation. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). pp. 4566-4575 (2015)",
+ "42. Wang, S., Yao, Z., Wang, R., Wu, Z., Chen, X.: FAIER: Fidelity and adequacy ensured image caption evaluation. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). pp. 14050-14059 (2021)",
+ "43. Zhang, T., Kishore, V., Wu, F., Weinberger, K.Q., Artzi, Y.: BERTScore: Evaluating text generation with bert. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020)",
+ "44. Zhao, T., Wu, X.: Pyramid feature attention network for saliency detection. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). pp. 3085-3094 (2019)"
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| Kosmos-2 | A man is walking an elephant on a leash in a park. | a large elephant (0.308) is walking through a park (0.516), with people walking behind it (0.454). The elephant (0.308) is walking on a wet sidewalk (0.858), ... In the background, there are trees (0.679) and a building, adding to the park atmosphere. \nUnmatched: building, atmosphere |
| CIDEr | SPICE | InfoMetric |
| 1.18 | 0.20 | 9.24 |
| CLIPScore | BLIPScore | HiFi-Score |
| 0.87 | 0.44 | 1.40 |
| InstructBUP | An elephant walking across a wet pavement in front of people. | In the image, a large elephant (0.308) is standing in the middle of a park (0.585), surrounded by people (0.523) ... The elephant (0.308) appears to be interacting with the people, possibly as part of an exhibit or show. A few benches (0.570) ... \nUnmatched: interacting, exhibit |
| CIDEr | SPICE | InfoMetric |
| 1.34 | 0.14 | 12.39 |
| CLIPScore | BLIPScore | HiFi-Score |
| 0.90 | 0.47 | 1.47 |
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+ "type": "ref_text",
+ "bbox": [
+ 0.226,
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+ "content": "4. Banerjee, S., Lavie, A.: METEOR: An automatic metric for mt evaluation with improved correlation with human judgments. In: Proceedings of the ACL Workshop on Intrinsic and Extrinsic Evaluation Measures for Machine Translation and/or Summarization. pp. 65-72 (2005)"
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+ "content": "0. Deng, C., Ding, N., Tan, M., Wu, Q.: Length-controllable image captioning. In: Proceedings of the European conference on computer vision (ECCV). pp. 712-729 (2020)"
+ },
+ {
+ "type": "ref_text",
+ "bbox": [
+ 0.226,
+ 0.786,
+ 0.786,
+ 0.84
+ ],
+ "angle": 0,
+ "content": "1. Deshpande, A., Aneja, J., Wang, L., Schwing, A.G., Forsyth, D.: Fast, diverse and accurate image captioning guided by part-of-speech. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). pp. 10695-10704 (2019)"
+ },
+ {
+ "type": "list",
+ "bbox": [
+ 0.226,
+ 0.266,
+ 0.786,
+ 0.84
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+ "angle": 0,
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+ "bbox": [
+ 0.218,
+ 0.147,
+ 0.787,
+ 0.204
+ ],
+ "angle": 0,
+ "content": "12. Devlin, J., Chang, M.W., Lee, K., Toutanova, K.: BERT: Pre-training of deep bidirectional transformers for language understanding. In: Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics (NAACL). pp. 4171-4186 (2019)"
+ },
+ {
+ "type": "ref_text",
+ "bbox": [
+ 0.218,
+ 0.205,
+ 0.787,
+ 0.26
+ ],
+ "angle": 0,
+ "content": "13. Hessel, J., Holtzman, A., Forbes, M., Le Bras, R., Choi, Y.: CLIPScore: A reference-free evaluation metric for image captioning. In: Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing (EMNLP). pp. 7514-7528 (2021)"
+ },
+ {
+ "type": "ref_text",
+ "bbox": [
+ 0.217,
+ 0.261,
+ 0.787,
+ 0.302
+ ],
+ "angle": 0,
+ "content": "14. Hodosh, M., Young, P., Hockenmaier, J.: Framing image description as a ranking task: Data, models and evaluation metrics. Journal of Artificial Intelligence Research 47, 853-899 (2013)"
+ },
+ {
+ "type": "ref_text",
+ "bbox": [
+ 0.217,
+ 0.304,
+ 0.787,
+ 0.346
+ ],
+ "angle": 0,
+ "content": "15. Hu, A., Chen, S., Zhang, L., Jin, Q.: InfoMetIC: An informative metric for reference-free image caption evaluation. In: Proceedings of the 61st Annual Meeting of the Association for Computational Linguistics (ACL). pp. 3171-3185 (2023)"
+ },
+ {
+ "type": "ref_text",
+ "bbox": [
+ 0.217,
+ 0.347,
+ 0.787,
+ 0.401
+ ],
+ "angle": 0,
+ "content": "16. Jain, J., Li, J., Chiu, M.T., Hassani, A., Orlov, N., Shi, H.: OneFormer: One transformer to rule universal image segmentation. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). pp. 2989-2998 (2023)"
+ },
+ {
+ "type": "ref_text",
+ "bbox": [
+ 0.217,
+ 0.403,
+ 0.787,
+ 0.472
+ ],
+ "angle": 0,
+ "content": "17. Jiang, M., Hu, J., Huang, Q., Zhang, L., Diesner, J., Gao, J.: REO-relevance, extraness, omission: A fine-grained evaluation for image captioning. In: Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP). pp. 1475-1480 (2019)"
+ },
+ {
+ "type": "ref_text",
+ "bbox": [
+ 0.217,
+ 0.474,
+ 0.787,
+ 0.543
+ ],
+ "angle": 0,
+ "content": "18. Jiang, M., Huang, Q., Zhang, L., Wang, X., Zhang, P., Gan, Z., Diesner, J., Gao, J.: TIGEr: Text-to-image grounding for image caption evaluation. In: Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP). pp. 2141-2152 (2019)"
+ },
+ {
+ "type": "ref_text",
+ "bbox": [
+ 0.217,
+ 0.545,
+ 0.787,
+ 0.586
+ ],
+ "angle": 0,
+ "content": "19. Johnson, J., Karpathy, A., Fei-Fei, L.: DenseCap: Fully convolutional localization networks for dense captioning. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). pp. 4565-4574 (2016)"
+ },
+ {
+ "type": "ref_text",
+ "bbox": [
+ 0.217,
+ 0.587,
+ 0.787,
+ 0.642
+ ],
+ "angle": 0,
+ "content": "20. Kasai, J., Sakaguchi, K., Dunagan, L., Morrison, J., Bras, R.L., Choi, Y., Smith, N.A.: Transparent human evaluation for image captioning. In: Proceedings of the 2022 Conference of the North American Chapter of the Association for Computational Linguistics (NAACL). pp. 3464-3478 (2022)"
+ },
+ {
+ "type": "ref_text",
+ "bbox": [
+ 0.217,
+ 0.643,
+ 0.787,
+ 0.698
+ ],
+ "angle": 0,
+ "content": "21. Kim, D.J., Choi, J., Oh, T.H., Kweon, I.S.: Dense relational captioning: Triple-stream networks for relationship-based captioning. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). pp. 6271-6280 (2019)"
+ },
+ {
+ "type": "ref_text",
+ "bbox": [
+ 0.217,
+ 0.7,
+ 0.787,
+ 0.74
+ ],
+ "angle": 0,
+ "content": "22. Kirillov, A., Mintun, E., Ravi, N., Mao, H., Rolland, C., Gustafson, L., Xiao, T., Whitehead, S., Berg, A.C., Lo, W.Y., et al.: Segment anything. arXiv preprint arXiv:2304.02643 (2023)"
+ },
+ {
+ "type": "ref_text",
+ "bbox": [
+ 0.217,
+ 0.742,
+ 0.787,
+ 0.784
+ ],
+ "angle": 0,
+ "content": "23. Krause, J., Johnson, J., Krishna, R., Fei-Fei, L.: A hierarchical approach for generating descriptive image paragraphs. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). pp. 317-325 (2017)"
+ },
+ {
+ "type": "ref_text",
+ "bbox": [
+ 0.217,
+ 0.785,
+ 0.787,
+ 0.84
+ ],
+ "angle": 0,
+ "content": "24. Krishna, R., Zhu, Y., Groth, O., Johnson, J., Hata, K., Kravitz, J., Chen, S., Kalantidis, Y., Li, L.J., Shamma, D.A., Bernstein, M.S., Fei-Fei, L.: Visual Genome: Connecting language and vision using crowdsourced dense image annotations. International Journal of Computer Vision 123, 32-73 (2017)"
+ },
+ {
+ "type": "list",
+ "bbox": [
+ 0.217,
+ 0.147,
+ 0.787,
+ 0.84
+ ],
+ "angle": 0,
+ "content": null
+ }
+ ],
+ [
+ {
+ "type": "header",
+ "bbox": [
+ 0.366,
+ 0.115,
+ 0.732,
+ 0.129
+ ],
+ "angle": 0,
+ "content": "HiFi-Score: Fine-grained Image Description Evaluation"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.769,
+ 0.117,
+ 0.785,
+ 0.127
+ ],
+ "angle": 0,
+ "content": "17"
+ },
+ {
+ "type": "ref_text",
+ "bbox": [
+ 0.217,
+ 0.147,
+ 0.786,
+ 0.189
+ ],
+ "angle": 0,
+ "content": "25. Kusner, M., Sun, Y., Kolkin, N., Weinberger, K.: From word embeddings to document distances. In: Proceedings of the 32nd International Conference on Machine Learning (ICML). pp. 957-966 (2015)"
+ },
+ {
+ "type": "ref_text",
+ "bbox": [
+ 0.217,
+ 0.19,
+ 0.787,
+ 0.259
+ ],
+ "angle": 0,
+ "content": "26. Lee, H., Yoon, S., Dernoncourt, F., Bui, T., Jung, K.: UMIC: An unreferenced metric for image captioning via contrastive learning. In: Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing (ACL-IJCNLP). pp. 220-226 (2021)"
+ },
+ {
+ "type": "ref_text",
+ "bbox": [
+ 0.217,
+ 0.26,
+ 0.786,
+ 0.3
+ ],
+ "angle": 0,
+ "content": "27. Lee, K.H., Chen, X., Hua, G., Hu, H., He, X.: Stacked cross attention for imagetext matching. In: Proceedings of the European conference on computer vision (ECCV). pp. 201-216 (2018)"
+ },
+ {
+ "type": "ref_text",
+ "bbox": [
+ 0.217,
+ 0.301,
+ 0.786,
+ 0.355
+ ],
+ "angle": 0,
+ "content": "28. Li, J., Li, D., Savarese, S., Hoi, S.: BLIP-2: bootstrapping language-image pretraining with frozen image encoders and large language models. In: Proceedings of the 40th International Conference on Machine Learning (ICML). pp. 19730-19742 (2023)"
+ },
+ {
+ "type": "ref_text",
+ "bbox": [
+ 0.217,
+ 0.356,
+ 0.786,
+ 0.412
+ ],
+ "angle": 0,
+ "content": "29. Li, L.H., Zhang, P., Zhang, H., Yang, J., Li, C., Zhong, Y., Wang, L., Yuan, L., Zhang, L., Hwang, J.N., et al.: Grounded language-image pre-training. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). pp. 10965-10975 (2022)"
+ },
+ {
+ "type": "ref_text",
+ "bbox": [
+ 0.217,
+ 0.412,
+ 0.786,
+ 0.439
+ ],
+ "angle": 0,
+ "content": "30. Lin, C.Y.: ROUGE: A package for automatic evaluation of summaries. In: Text Summarization Branches Out. pp. 74-81 (2004)"
+ },
+ {
+ "type": "ref_text",
+ "bbox": [
+ 0.217,
+ 0.44,
+ 0.786,
+ 0.481
+ ],
+ "angle": 0,
+ "content": "31. Liu, H., Li, C., Wu, Q., Lee, Y.J.: Visual instruction tuning. In: Proceedings of the 37th International Conference on Neural Information Processing Systems (NeurIPS). pp. 34892-34916 (2023)"
+ },
+ {
+ "type": "ref_text",
+ "bbox": [
+ 0.217,
+ 0.481,
+ 0.786,
+ 0.536
+ ],
+ "angle": 0,
+ "content": "32. Lu, Y., Yang, X., Li, X., Wang, X.E., Wang, W.Y.: LLMScore: Unveiling the power of large language models in text-to-image synthesis evaluation. In: Proceedings of the 37th International Conference on Neural Information Processing Systems (NeurIPS). pp. 23075-23093 (2023)"
+ },
+ {
+ "type": "ref_text",
+ "bbox": [
+ 0.217,
+ 0.536,
+ 0.786,
+ 0.578
+ ],
+ "angle": 0,
+ "content": "33. Madhyastha, P., Wang, J., Specia, L.: VIFIDEL: Evaluating the visual fidelity of image descriptions. In: Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics (ACL). pp. 6539-6550 (2019)"
+ },
+ {
+ "type": "ref_text",
+ "bbox": [
+ 0.217,
+ 0.578,
+ 0.786,
+ 0.605
+ ],
+ "angle": 0,
+ "content": "34. Navon, D.: Forest before trees: The precedence of global features in visual perception. Cognitive Psychology 9(3), 353-383 (1977)"
+ },
+ {
+ "type": "ref_text",
+ "bbox": [
+ 0.217,
+ 0.606,
+ 0.786,
+ 0.647
+ ],
+ "angle": 0,
+ "content": "35. Papineni, K., Roukos, S., Ward, T., Zhu, W.J.: BLEU: A method for automatic evaluation of machine translation. In: Proceedings of the 40th Annual Meeting of the Association for Computational Linguistics (ACL). pp. 311-318 (2002)"
+ },
+ {
+ "type": "ref_text",
+ "bbox": [
+ 0.217,
+ 0.647,
+ 0.786,
+ 0.689
+ ],
+ "angle": 0,
+ "content": "36. Peng, Z., Wang, W., Dong, L., Hao, Y., Huang, S., Ma, S., Ye, Q., Wei, F.: Grounding multimodal large language models to the world. In: Proceedings of the 12th International Conference on Learning Representations (ICLR) (2024)"
+ },
+ {
+ "type": "ref_text",
+ "bbox": [
+ 0.217,
+ 0.689,
+ 0.786,
+ 0.73
+ ],
+ "angle": 0,
+ "content": "37. Pont-Tuset, J., Uijlings, J., Changpinyo, S., Soricut, R., Ferrari, V.: Connecting vision and language with localized narratives. In: Proceedings of the European conference on computer vision (ECCV). pp. 647-664 (2020)"
+ },
+ {
+ "type": "ref_text",
+ "bbox": [
+ 0.217,
+ 0.73,
+ 0.786,
+ 0.785
+ ],
+ "angle": 0,
+ "content": "38. Radford, A., Kim, J.W., Hallacy, C., Ramesh, A., Goh, G., Agarwal, S., Sastry, G., Askell, A., Mishkin, P., Clark, J., et al.: Learning transferable visual models from natural language supervision. In: Proceedings of the 38th International Conference on Machine Learning. pp. 8748-8763 (2021)"
+ },
+ {
+ "type": "ref_text",
+ "bbox": [
+ 0.217,
+ 0.785,
+ 0.786,
+ 0.841
+ ],
+ "angle": 0,
+ "content": "39. Sarto, S., Barraco, M., Cornia, M., Baraldi, L., Cucchiara, R.: Positive-augmented contrastive learning for image and video captioning evaluation. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). pp. 6914-6924 (2023)"
+ },
+ {
+ "type": "list",
+ "bbox": [
+ 0.217,
+ 0.147,
+ 0.787,
+ 0.841
+ ],
+ "angle": 0,
+ "content": null
+ }
+ ],
+ [
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.218,
+ 0.116,
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+ "angle": 0,
+ "content": "18"
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+ {
+ "type": "header",
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+ },
+ {
+ "type": "ref_text",
+ "bbox": [
+ 0.218,
+ 0.147,
+ 0.787,
+ 0.189
+ ],
+ "angle": 0,
+ "content": "40. Sharif, N., Bennamoun, M., White, L.R., Shah, S.A.A.: Learning-based composite metrics for improved caption evaluation. In: Proceedings of ACL 2018, Student Research Workshop. pp. 14-20 (2018)"
+ },
+ {
+ "type": "ref_text",
+ "bbox": [
+ 0.217,
+ 0.19,
+ 0.787,
+ 0.232
+ ],
+ "angle": 0,
+ "content": "41. Vedantam, R., Lawrence Zitnick, C., Parikh, D.: CIDEr: Consensus-based image description evaluation. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). pp. 4566-4575 (2015)"
+ },
+ {
+ "type": "ref_text",
+ "bbox": [
+ 0.217,
+ 0.232,
+ 0.787,
+ 0.273
+ ],
+ "angle": 0,
+ "content": "42. Wang, S., Yao, Z., Wang, R., Wu, Z., Chen, X.: FAIER: Fidelity and adequacy ensured image caption evaluation. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). pp. 14050-14059 (2021)"
+ },
+ {
+ "type": "ref_text",
+ "bbox": [
+ 0.217,
+ 0.273,
+ 0.787,
+ 0.314
+ ],
+ "angle": 0,
+ "content": "43. Zhang, T., Kishore, V., Wu, F., Weinberger, K.Q., Artzi, Y.: BERTScore: Evaluating text generation with bert. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020)"
+ },
+ {
+ "type": "ref_text",
+ "bbox": [
+ 0.217,
+ 0.315,
+ 0.787,
+ 0.356
+ ],
+ "angle": 0,
+ "content": "44. Zhao, T., Wu, X.: Pyramid feature attention network for saliency detection. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). pp. 3085-3094 (2019)"
+ },
+ {
+ "type": "list",
+ "bbox": [
+ 0.217,
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+# HiFi-Score: Fine-grained Image Description Evaluation with Hierarchical Parsing Graphs
+
+Ziwei Yao $^{1,2}$ , Ruiping Wang $^{1,2}$ , and Xilin Chen $^{1,2}$
+
+ | Caption | Paragraph |
| Kosmos-2 | A man is walking an elephant on a leash in a park. | a large elephant (0.308) is walking through a park (0.516), with people walking behind it (0.454). The elephant (0.308) is walking on a wet sidewalk (0.858), ... In the background, there are trees (0.679) and a building, adding to the park atmosphere.
+Unmatched: building, atmosphere |
| CIDEr | SPICE | InfoMetric |
| 1.18 | 0.20 | 9.24 |
| CLIPScore | BLIPScore | HiFi-Score |
| 0.87 | 0.44 | 1.40 |
| InstructBUP | An elephant walking across a wet pavement in front of people. | In the image, a large elephant (0.308) is standing in the middle of a park (0.585), surrounded by people (0.523) ... The elephant (0.308) appears to be interacting with the people, possibly as part of an exhibit or show. A few benches (0.570) ...
+Unmatched: interacting, exhibit |
| CIDEr | SPICE | InfoMetric |
| 1.34 | 0.14 | 12.39 |
| CLIPScore | BLIPScore | HiFi-Score |
| 0.90 | 0.47 | 1.47 |
+
+# 5 Conclusion
+
+In conclusion, we propose a novel image description evaluation metric, HiFi-Score, which provides fine-grained multi-dimensional evaluation based on hierarchical parsing graphs. Specifically, we model both images and texts as HPGs, analyzing all semantic instances from global to local and incorporating attributes and semantic relationships. Then, the scores of fidelity and adequacy are calculated gradually from the bottom up. We further evaluate the fluency of language expression taking the advantage of the LLMs. Experimental results on caption-level datasets demonstrate HiFi-Score's higher human consistency. Moreover, we construct paragraph-level dataset ParaEval to evaluate current metrics with long-context descriptions, and validate the superiority of HiFi-Score. Extensive evaluations demonstrate the potential of HiFi-Score for evaluating VLMs' generative capabilities and the diverse image description tasks.
+
+# Acknowledgements
+
+This work is partially supported by Natural Science Foundation of China under contract No. U21B2025, and National Key R&D Program of China No. 2021ZD0111901, 2023YFF1105104.
+
+# References
+
+1. Aditya, S., Yang, Y., Baral, C., Aloimonos, Y., Fermüller, C.: Image understanding using vision and reasoning through scene description graph. Computer Vision and Image Understanding 173, 33-45 (2018)
+2. Alayrac, J.B., Donahue, J., Luc, P., Miech, A., Barr, I., Hasson, Y., Lenc, K., Mensch, A., Millican, K., Reynolds, M., Ring, R., Rutherford, E., Cabi, S., Han, T., Gong, Z., Samangooei, S., Monteiro, M., Menick, J., Borgeaud, S., Brock, A., Nematzadeh, A., Sharifzadeh, S., Binkowski, M., Barreira, R., Vinyals, O., Zisserman, A., Simonyan, K.: Flamingo: a visual language model for few-shot learning. In: Proceedings of the 36th International Conference on Neural Information Processing Systems (NeurIPS). pp. 23716-23736 (2022)
+3. Anderson, P., Fernando, B., Johnson, M., Gould, S.: SPICE: Semantic propositional image caption evaluation. In: Proceedings of the European conference on computer vision (ECCV). pp. 382-398 (2016)
+4. Banerjee, S., Lavie, A.: METEOR: An automatic metric for mt evaluation with improved correlation with human judgments. In: Proceedings of the ACL Workshop on Intrinsic and Extrinsic Evaluation Measures for Machine Translation and/or Summarization. pp. 65-72 (2005)
+5. Chan, D., Petryk, S., Gonzalez, J., Darrell, T., Canny, J.: CLAIR: Evaluating image captions with large language models. In: Proceedings of the 2023 Conference on Empirical Methods in Natural Language Processing (EMNLP). pp. 13638-13646 (2023)
+6. Chen, Y.C., Li, L., Yu, L., El Kholy, A., Ahmed, F., Gan, Z., Cheng, Y., Liu, J.: UNITER: Universal image-text representation learning. In: Proceedings of the European conference on computer vision (ECCV). pp. 104-120 (2020)
+7. Cornia, M., Baraldi, L., Cucchiara, R.: Show, control and tell: A framework for generating controllable and grounded captions. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). pp. 8307-8316 (2019)
+8. Cui, Y., Yang, G., Veit, A., Huang, X., Belongie, S.: Learning to evaluate image captioning. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). pp. 5804-5812 (2018)
+9. Dai, W., Li, J., Li, D., Tiong, A., Zhao, J., Wang, W., Li, B., Fung, P., Hoi, S.: InstructBLIP: Towards general-purpose vision-language models with instruction tuning. In: Proceedings of the 37th International Conference on Neural Information Processing Systems (NeurIPS). pp. 49250-49267 (2023)
+0. Deng, C., Ding, N., Tan, M., Wu, Q.: Length-controllable image captioning. In: Proceedings of the European conference on computer vision (ECCV). pp. 712-729 (2020)
+1. Deshpande, A., Aneja, J., Wang, L., Schwing, A.G., Forsyth, D.: Fast, diverse and accurate image captioning guided by part-of-speech. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). pp. 10695-10704 (2019)
+
+12. Devlin, J., Chang, M.W., Lee, K., Toutanova, K.: BERT: Pre-training of deep bidirectional transformers for language understanding. In: Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics (NAACL). pp. 4171-4186 (2019)
+13. Hessel, J., Holtzman, A., Forbes, M., Le Bras, R., Choi, Y.: CLIPScore: A reference-free evaluation metric for image captioning. In: Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing (EMNLP). pp. 7514-7528 (2021)
+14. Hodosh, M., Young, P., Hockenmaier, J.: Framing image description as a ranking task: Data, models and evaluation metrics. Journal of Artificial Intelligence Research 47, 853-899 (2013)
+15. Hu, A., Chen, S., Zhang, L., Jin, Q.: InfoMetIC: An informative metric for reference-free image caption evaluation. In: Proceedings of the 61st Annual Meeting of the Association for Computational Linguistics (ACL). pp. 3171-3185 (2023)
+16. Jain, J., Li, J., Chiu, M.T., Hassani, A., Orlov, N., Shi, H.: OneFormer: One transformer to rule universal image segmentation. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). pp. 2989-2998 (2023)
+17. Jiang, M., Hu, J., Huang, Q., Zhang, L., Diesner, J., Gao, J.: REO-relevance, extraness, omission: A fine-grained evaluation for image captioning. In: Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP). pp. 1475-1480 (2019)
+18. Jiang, M., Huang, Q., Zhang, L., Wang, X., Zhang, P., Gan, Z., Diesner, J., Gao, J.: TIGEr: Text-to-image grounding for image caption evaluation. In: Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP). pp. 2141-2152 (2019)
+19. Johnson, J., Karpathy, A., Fei-Fei, L.: DenseCap: Fully convolutional localization networks for dense captioning. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). pp. 4565-4574 (2016)
+20. Kasai, J., Sakaguchi, K., Dunagan, L., Morrison, J., Bras, R.L., Choi, Y., Smith, N.A.: Transparent human evaluation for image captioning. In: Proceedings of the 2022 Conference of the North American Chapter of the Association for Computational Linguistics (NAACL). pp. 3464-3478 (2022)
+21. Kim, D.J., Choi, J., Oh, T.H., Kweon, I.S.: Dense relational captioning: Triple-stream networks for relationship-based captioning. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). pp. 6271-6280 (2019)
+22. Kirillov, A., Mintun, E., Ravi, N., Mao, H., Rolland, C., Gustafson, L., Xiao, T., Whitehead, S., Berg, A.C., Lo, W.Y., et al.: Segment anything. arXiv preprint arXiv:2304.02643 (2023)
+23. Krause, J., Johnson, J., Krishna, R., Fei-Fei, L.: A hierarchical approach for generating descriptive image paragraphs. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). pp. 317-325 (2017)
+24. Krishna, R., Zhu, Y., Groth, O., Johnson, J., Hata, K., Kravitz, J., Chen, S., Kalantidis, Y., Li, L.J., Shamma, D.A., Bernstein, M.S., Fei-Fei, L.: Visual Genome: Connecting language and vision using crowdsourced dense image annotations. International Journal of Computer Vision 123, 32-73 (2017)
+
+25. Kusner, M., Sun, Y., Kolkin, N., Weinberger, K.: From word embeddings to document distances. In: Proceedings of the 32nd International Conference on Machine Learning (ICML). pp. 957-966 (2015)
+26. Lee, H., Yoon, S., Dernoncourt, F., Bui, T., Jung, K.: UMIC: An unreferenced metric for image captioning via contrastive learning. In: Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing (ACL-IJCNLP). pp. 220-226 (2021)
+27. Lee, K.H., Chen, X., Hua, G., Hu, H., He, X.: Stacked cross attention for imagetext matching. In: Proceedings of the European conference on computer vision (ECCV). pp. 201-216 (2018)
+28. Li, J., Li, D., Savarese, S., Hoi, S.: BLIP-2: bootstrapping language-image pretraining with frozen image encoders and large language models. In: Proceedings of the 40th International Conference on Machine Learning (ICML). pp. 19730-19742 (2023)
+29. Li, L.H., Zhang, P., Zhang, H., Yang, J., Li, C., Zhong, Y., Wang, L., Yuan, L., Zhang, L., Hwang, J.N., et al.: Grounded language-image pre-training. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). pp. 10965-10975 (2022)
+30. Lin, C.Y.: ROUGE: A package for automatic evaluation of summaries. In: Text Summarization Branches Out. pp. 74-81 (2004)
+31. Liu, H., Li, C., Wu, Q., Lee, Y.J.: Visual instruction tuning. In: Proceedings of the 37th International Conference on Neural Information Processing Systems (NeurIPS). pp. 34892-34916 (2023)
+32. Lu, Y., Yang, X., Li, X., Wang, X.E., Wang, W.Y.: LLMScore: Unveiling the power of large language models in text-to-image synthesis evaluation. In: Proceedings of the 37th International Conference on Neural Information Processing Systems (NeurIPS). pp. 23075-23093 (2023)
+33. Madhyastha, P., Wang, J., Specia, L.: VIFIDEL: Evaluating the visual fidelity of image descriptions. In: Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics (ACL). pp. 6539-6550 (2019)
+34. Navon, D.: Forest before trees: The precedence of global features in visual perception. Cognitive Psychology 9(3), 353-383 (1977)
+35. Papineni, K., Roukos, S., Ward, T., Zhu, W.J.: BLEU: A method for automatic evaluation of machine translation. In: Proceedings of the 40th Annual Meeting of the Association for Computational Linguistics (ACL). pp. 311-318 (2002)
+36. Peng, Z., Wang, W., Dong, L., Hao, Y., Huang, S., Ma, S., Ye, Q., Wei, F.: Grounding multimodal large language models to the world. In: Proceedings of the 12th International Conference on Learning Representations (ICLR) (2024)
+37. Pont-Tuset, J., Uijlings, J., Changpinyo, S., Soricut, R., Ferrari, V.: Connecting vision and language with localized narratives. In: Proceedings of the European conference on computer vision (ECCV). pp. 647-664 (2020)
+38. Radford, A., Kim, J.W., Hallacy, C., Ramesh, A., Goh, G., Agarwal, S., Sastry, G., Askell, A., Mishkin, P., Clark, J., et al.: Learning transferable visual models from natural language supervision. In: Proceedings of the 38th International Conference on Machine Learning. pp. 8748-8763 (2021)
+39. Sarto, S., Barraco, M., Cornia, M., Baraldi, L., Cucchiara, R.: Positive-augmented contrastive learning for image and video captioning evaluation. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). pp. 6914-6924 (2023)
+
+40. Sharif, N., Bennamoun, M., White, L.R., Shah, S.A.A.: Learning-based composite metrics for improved caption evaluation. In: Proceedings of ACL 2018, Student Research Workshop. pp. 14-20 (2018)
+41. Vedantam, R., Lawrence Zitnick, C., Parikh, D.: CIDEr: Consensus-based image description evaluation. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). pp. 4566-4575 (2015)
+42. Wang, S., Yao, Z., Wang, R., Wu, Z., Chen, X.: FAIER: Fidelity and adequacy ensured image caption evaluation. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). pp. 14050-14059 (2021)
+43. Zhang, T., Kishore, V., Wu, F., Weinberger, K.Q., Artzi, Y.: BERTScore: Evaluating text generation with bert. In: Proceedings of the 8th International Conference on Learning Representations (ICLR) (2020)
+44. Zhao, T., Wu, X.: Pyramid feature attention network for saliency detection. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). pp. 3085-3094 (2019)
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+ 138,
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+ 243
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+ "content": "1. Aditya, S., Yang, Y., Baral, C., Aloimonos, Y., Fermüller, C.: Image understanding using vision and reasoning through scene description graph. Computer Vision and Image Understanding 173, 33-45 (2018)"
+ }
+ ]
+ }
+ ],
+ "index": 5
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+ {
+ "bbox": [
+ 138,
+ 244,
+ 481,
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+ 138,
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+ {
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+ 138,
+ 244,
+ 481,
+ 319
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+ {
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+ 138,
+ 320,
+ 481,
+ 352
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+ "type": "text",
+ "content": "3. Anderson, P., Fernando, B., Johnson, M., Gould, S.: SPICE: Semantic propositional image caption evaluation. In: Proceedings of the European conference on computer vision (ECCV). pp. 382-398 (2016)"
+ }
+ ]
+ }
+ ],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 138,
+ 353,
+ 481,
+ 396
+ ],
+ "type": "ref_text",
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+ "bbox": [
+ 138,
+ 353,
+ 481,
+ 396
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 138,
+ 353,
+ 481,
+ 396
+ ],
+ "type": "text",
+ "content": "4. Banerjee, S., Lavie, A.: METEOR: An automatic metric for mt evaluation with improved correlation with human judgments. In: Proceedings of the ACL Workshop on Intrinsic and Extrinsic Evaluation Measures for Machine Translation and/or Summarization. pp. 65-72 (2005)"
+ }
+ ]
+ }
+ ],
+ "index": 8
+ },
+ {
+ "bbox": [
+ 138,
+ 396,
+ 481,
+ 438
+ ],
+ "type": "ref_text",
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+ {
+ "bbox": [
+ 138,
+ 396,
+ 481,
+ 438
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 138,
+ 396,
+ 481,
+ 438
+ ],
+ "type": "text",
+ "content": "5. Chan, D., Petryk, S., Gonzalez, J., Darrell, T., Canny, J.: CLAIR: Evaluating image captions with large language models. In: Proceedings of the 2023 Conference on Empirical Methods in Natural Language Processing (EMNLP). pp. 13638-13646 (2023)"
+ }
+ ]
+ }
+ ],
+ "index": 9
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+ {
+ "bbox": [
+ 138,
+ 438,
+ 481,
+ 471
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+ 138,
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+ 481,
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+ ],
+ "spans": [
+ {
+ "bbox": [
+ 138,
+ 438,
+ 481,
+ 471
+ ],
+ "type": "text",
+ "content": "6. Chen, Y.C., Li, L., Yu, L., El Kholy, A., Ahmed, F., Gan, Z., Cheng, Y., Liu, J.: UNITER: Universal image-text representation learning. In: Proceedings of the European conference on computer vision (ECCV). pp. 104-120 (2020)"
+ }
+ ]
+ }
+ ],
+ "index": 10
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+ {
+ "bbox": [
+ 138,
+ 471,
+ 481,
+ 514
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+ "type": "ref_text",
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+ "bbox": [
+ 138,
+ 471,
+ 481,
+ 514
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 138,
+ 471,
+ 481,
+ 514
+ ],
+ "type": "text",
+ "content": "7. Cornia, M., Baraldi, L., Cucchiara, R.: Show, control and tell: A framework for generating controllable and grounded captions. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). pp. 8307-8316 (2019)"
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+ {
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+ 138,
+ 514,
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+ 132,
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+ }
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+ "index": 1
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+ "page_size": [
+ 612,
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+ "page_idx": 15
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+ {
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+ 132,
+ 116,
+ 481,
+ 149
+ ],
+ "type": "text",
+ "content": "25. Kusner, M., Sun, Y., Kolkin, N., Weinberger, K.: From word embeddings to document distances. In: Proceedings of the 32nd International Conference on Machine Learning (ICML). pp. 957-966 (2015)"
+ }
+ ]
+ }
+ ],
+ "index": 2
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+ {
+ "bbox": [
+ 132,
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+ {
+ "bbox": [
+ 132,
+ 150,
+ 481,
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+ "type": "text",
+ "content": "26. Lee, H., Yoon, S., Dernoncourt, F., Bui, T., Jung, K.: UMIC: An unreferenced metric for image captioning via contrastive learning. In: Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing (ACL-IJCNLP). pp. 220-226 (2021)"
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+ 132,
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+ {
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+ 132,
+ 205,
+ 481,
+ 237
+ ],
+ "type": "text",
+ "content": "27. Lee, K.H., Chen, X., Hua, G., Hu, H., He, X.: Stacked cross attention for imagetext matching. In: Proceedings of the European conference on computer vision (ECCV). pp. 201-216 (2018)"
+ }
+ ]
+ }
+ ],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 132,
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+ {
+ "bbox": [
+ 132,
+ 238,
+ 481,
+ 281
+ ],
+ "type": "text",
+ "content": "28. Li, J., Li, D., Savarese, S., Hoi, S.: BLIP-2: bootstrapping language-image pretraining with frozen image encoders and large language models. In: Proceedings of the 40th International Conference on Machine Learning (ICML). pp. 19730-19742 (2023)"
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+ ],
+ "index": 5
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+ {
+ "bbox": [
+ 132,
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+ "spans": [
+ {
+ "bbox": [
+ 132,
+ 281,
+ 481,
+ 326
+ ],
+ "type": "text",
+ "content": "29. Li, L.H., Zhang, P., Zhang, H., Yang, J., Li, C., Zhong, Y., Wang, L., Yuan, L., Zhang, L., Hwang, J.N., et al.: Grounded language-image pre-training. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). pp. 10965-10975 (2022)"
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+ ],
+ "index": 6
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+ {
+ "bbox": [
+ 132,
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+ {
+ "bbox": [
+ 132,
+ 326,
+ 481,
+ 347
+ ],
+ "type": "text",
+ "content": "30. Lin, C.Y.: ROUGE: A package for automatic evaluation of summaries. In: Text Summarization Branches Out. pp. 74-81 (2004)"
+ }
+ ]
+ }
+ ],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 132,
+ 348,
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+ 132,
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+ ],
+ "spans": [
+ {
+ "bbox": [
+ 132,
+ 348,
+ 481,
+ 380
+ ],
+ "type": "text",
+ "content": "31. Liu, H., Li, C., Wu, Q., Lee, Y.J.: Visual instruction tuning. In: Proceedings of the 37th International Conference on Neural Information Processing Systems (NeurIPS). pp. 34892-34916 (2023)"
+ }
+ ]
+ }
+ ],
+ "index": 8
+ },
+ {
+ "bbox": [
+ 132,
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+ ],
+ "spans": [
+ {
+ "bbox": [
+ 132,
+ 380,
+ 481,
+ 424
+ ],
+ "type": "text",
+ "content": "32. Lu, Y., Yang, X., Li, X., Wang, X.E., Wang, W.Y.: LLMScore: Unveiling the power of large language models in text-to-image synthesis evaluation. In: Proceedings of the 37th International Conference on Neural Information Processing Systems (NeurIPS). pp. 23075-23093 (2023)"
+ }
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+ }
+ ],
+ "index": 9
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+ {
+ "bbox": [
+ 132,
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+ "type": "ref_text",
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+ 132,
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+ ],
+ "spans": [
+ {
+ "bbox": [
+ 132,
+ 424,
+ 481,
+ 457
+ ],
+ "type": "text",
+ "content": "33. Madhyastha, P., Wang, J., Specia, L.: VIFIDEL: Evaluating the visual fidelity of image descriptions. In: Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics (ACL). pp. 6539-6550 (2019)"
+ }
+ ]
+ }
+ ],
+ "index": 10
+ },
+ {
+ "bbox": [
+ 132,
+ 457,
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+ "type": "ref_text",
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+ 132,
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+ ],
+ "spans": [
+ {
+ "bbox": [
+ 132,
+ 457,
+ 481,
+ 479
+ ],
+ "type": "text",
+ "content": "34. Navon, D.: Forest before trees: The precedence of global features in visual perception. Cognitive Psychology 9(3), 353-383 (1977)"
+ }
+ ]
+ }
+ ],
+ "index": 11
+ },
+ {
+ "bbox": [
+ 132,
+ 479,
+ 481,
+ 512
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+ "type": "ref_text",
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+ {
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+ ],
+ "spans": [
+ {
+ "bbox": [
+ 132,
+ 479,
+ 481,
+ 512
+ ],
+ "type": "text",
+ "content": "35. Papineni, K., Roukos, S., Ward, T., Zhu, W.J.: BLEU: A method for automatic evaluation of machine translation. In: Proceedings of the 40th Annual Meeting of the Association for Computational Linguistics (ACL). pp. 311-318 (2002)"
+ }
+ ]
+ }
+ ],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 132,
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+ 481,
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+ "type": "ref_text",
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+ "spans": [
+ {
+ "bbox": [
+ 132,
+ 512,
+ 481,
+ 545
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+ "type": "text",
+ "content": "36. Peng, Z., Wang, W., Dong, L., Hao, Y., Huang, S., Ma, S., Ye, Q., Wei, F.: Grounding multimodal large language models to the world. In: Proceedings of the 12th International Conference on Learning Representations (ICLR) (2024)"
+ }
+ ]
+ }
+ ],
+ "index": 13
+ },
+ {
+ "bbox": [
+ 132,
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+ {
+ "bbox": [
+ 132,
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+ 481,
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+ "type": "text",
+ "content": "37. Pont-Tuset, J., Uijlings, J., Changpinyo, S., Soricut, R., Ferrari, V.: Connecting vision and language with localized narratives. In: Proceedings of the European conference on computer vision (ECCV). pp. 647-664 (2020)"
+ }
+ ]
+ }
+ ],
+ "index": 14
+ },
+ {
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+ 132,
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+ {
+ "bbox": [
+ 132,
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+ "type": "text",
+ "content": "38. Radford, A., Kim, J.W., Hallacy, C., Ramesh, A., Goh, G., Agarwal, S., Sastry, G., Askell, A., Mishkin, P., Clark, J., et al.: Learning transferable visual models from natural language supervision. In: Proceedings of the 38th International Conference on Machine Learning. pp. 8748-8763 (2021)"
+ }
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+ }
+ ],
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diff --git a/2024/HiT-SR_ Hierarchical Transformer for Efficient Image Super-Resolution/8da60c30-c0b4-4adb-9b08-260cf5e0cb32_content_list.json b/2024/HiT-SR_ Hierarchical Transformer for Efficient Image Super-Resolution/8da60c30-c0b4-4adb-9b08-260cf5e0cb32_content_list.json
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+ "text": "HiT-SR: Hierarchical Transformer for Efficient Image Super-Resolution",
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+ "text": "Xiang Zhang $^{1}$ , Yulun Zhang $^{2*}$ , and Fisher $\\mathrm{Yu}^{1}$",
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+ "text": "$^{1}$ ETH Zürich, Switzerland \n $^{2}$ MoE Key Lab of Artificial Intelligence, Shanghai Jiao Tong University, China {xiangz.ethz,yulun100}@gmail.com, i@yf.io https://github.com/XiangZ-0/HiT-SR",
+ "bbox": [
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+ "page_idx": 0
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+ {
+ "type": "text",
+ "text": "Abstract. Transformers have exhibited promising performance in computer vision tasks including image super-resolution (SR). However, popular transformer-based SR methods often employ window self-attention with quadratic computational complexity to window sizes, resulting in fixed small windows with limited receptive fields. In this paper, we present a general strategy to convert transformer-based SR networks to hierarchical transformers (HiT-SR), boosting SR performance with multi-scale features while maintaining an efficient design. Specifically, we first replace the commonly used fixed small windows with expanding hierarchical windows to aggregate features at different scales and establish long-range dependencies. Considering the intensive computation required for large windows, we further design a spatial-channel correlation method with linear complexity to window sizes, efficiently gathering spatial and channel information from hierarchical windows. Extensive experiments verify the effectiveness and efficiency of our HiT-SR, and our improved versions of SwinIR-Light, SwinIR-NG, and SRFormer-Light yield state-of-the-art SR results with fewer parameters, FLOPs, and faster speeds $(\\sim 7\\times)$ .",
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+ "text": "Keywords: Super-Resolution $\\cdot$ Transformer $\\cdot$ Hierarchical Windows",
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+ "text": "1 Introduction",
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+ {
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+ "text": "Image super-resolution (SR) is a classical low-level vision task that aims to convert a low-resolution (LR) image to its high-resolution (HR) counterpart with better visual details. How to tackle the ill-posed SR problem has attracted considerable interest for decades [13, 17, 18, 20, 46, 47]. Many popular approaches employ convolutional neural networks (CNNs) to learn the projection between LR inputs and HR images [11, 13, 34, 35, 51, 52]. Despite significant progress being achieved, CNN-based methods usually focus on utilizing local features via convolution and often fall short in aggregating long-range information across the image, limiting the performance of CNN-based SR.",
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+ "text": "The recent development of vision transformers provides a promising solution for establishing long-range dependencies [8, 12, 14, 30, 42], benefiting many",
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+ "text": "* Corresponding author: Yulun Zhang.",
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+ "(b) Complexity comparisons"
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+ "Fig. 1: Comparisons of the popular efficient SR transformers, i.e., SwinIR-Light (SIR) [26], SwinIR-NG (SNG) [10], and SRFormer-Light (SRF) [55], and the corresponding HiT-SR versions, i.e., HiT-SIR, HiT-SNG, and HiT-SRF. The complexity metrics are calculated under $\\times 2$ upscaling on an A100 GPU, with the output size set to $720 \\times 1280$ ."
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+ "text": "computer vision tasks including image SR [9, 10, 16, 26, 50]. An essential component in popular transformer-based SR methods is the window self-attention (W-SA) [10, 26, 29, 30]. By bringing locality into self-attention, the W-SA mechanism not only better utilizes spatial information from input images but also mitigates the computational burden when processing high-resolution images [26, 30]. However, current transformer-based SR methods often employ W-SA with fixed small window sizes, e.g., $8 \\times 8$ in SwinIR [26], limiting the receptive field to a single scale and preventing the network from gathering multi-scale information such as local textures and repetitive patterns [22, 23, 25, 27]. In addition, the quadratic computational complexity of W-SA to the window size also makes the expansion of receptive fields unaffordable in practice. To mitigate the computational overhead, previous attempts often reduce channels to support large windows, e.g., channel splitting of group-wise multi-scale self-attention (GMSA) in ELAN [50] and channel compression of permuted self-attention block (PSA) in SRFormer [55]. However, these methods not only suffer from the trade-off between spatial and channel information but also remain quadratic complexity to window sizes, limiting the window scaling (max $16 \\times 16$ in ELAN [50] and $24 \\times 24$ in SRFormer [55] vs. $64 \\times 64$ and larger in ours). Therefore, how to effectively aggregate multi-scale features while maintaining computational efficiency remains a critical problem for transformer-based SR approaches.",
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+ "text": "To this end, we develop a general strategy to convert popular transformer-based SR networks to hierarchical transformers for efficient image SR (HiT-SR). Motivated by the success of multi-scale feature aggregation in SR [22, 23, 25, 50], we first propose to replace the fixed small windows with expanding hierarchical windows in transformer layers, enabling HiT-SR to leverage informative multiscale features with gradually enlarging receptive fields. To cope with the increasing computational burdens of W-SA in handling large windows, we further",
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+ {
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+ "text": "design a spatial-channel correlation (SCC) method for efficient aggregation of hierarchical features. Specifically, our SCC consists of a dual feature extraction (DFE) layer to improve feature projection by combining spatial and channel information, a spatial and channel self-correlation (S-SC and C-SC) approach to efficiently exploit hierarchical features with linear computational complexity to window sizes, better supporting window scaling. In addition, unlike the conventional W-SA that employs hardware inefficient softmax layers [5] and time-consuming window shifting operations, our SCC directly uses feature correlation matrices for transformation and hierarchical windows for receptive field expansion, boosting computational efficiency while preserving functionality.",
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+ "- We propose a simple yet effective strategy, i.e., HiT-SR, to convert popular transformer-based SR methods to our hierarchical transformers, boosting SR performance by exploiting multi-scale features and long-range dependencies.",
+ "- We design a spatial-channel correlation method to efficiently leverage spatial and channel features with linear computational complexity to window sizes, enabling utilization of large hierarchical windows, e.g., $64 \\times 64$ windows.",
+ "- We convert SwinIR-Light [26], SwinIR-NG [10] and SRFormer-Light [55] to HiT-SR versions, i.e., HiT-SIR, HiT-SNG, and HiT-SRF, achieving better performance with fewer parameters, FLOPs, and $\\sim 7\\times$ speed-up (Fig. 1)."
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+ "text": "Efficient SR. Several CNN-based methods are proposed to approach SR in an efficient way. Previous works first explore compact building blocks for SR networks [2,15,27]. Meanwhile, information distillation methods are developed to progressively refine image features for better performance [22,28], and LatticeNet designs an economical SR structure based on the lattice filter bank [31]. Recent methods further improve the efficiency of image SR by utilizing network pruning techniques [41,53]. However, most CNN-based SR approaches focus on exploiting local features and struggle to utilize long-range dependencies in image SR.",
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+ "text": "Transformer-based SR. The self-attention (SA) mechanism has been widely employed to establish long-range dependencies in both high-level [6, 14, 29, 30, 37, 38, 54] and low-level vision tasks [7, 26, 44, 48]. For the SR task, SwinIR [26] designs a general image restoration framework based on the shifted window self-attention (SW-SA) [30], and several techniques have been developed to utilize a wider range of features, including N-Gram [10], omni self-attention (OSA) [40], and permuted self-attention (PSA) [55]. Although many advances have been made, most existing works neglect the importance of hierarchical features in SR due to the intensive computation required by W-SA on large windows.",
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+ "text": "Hierarchical Feature. Hierarchical features have been proven effective in boosting SR performance [22,23,25,27,50], but it is generally challenging for transformer-based SR methods to utilize hierarchical features due to the quadratic complexity of W-SA to window sizes. Recent method ELAN designs a group-wise multiscale SA (GMSA) technique to gather features at different scales [50]. However,",
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+ "image_caption": [
+ "Fig. 2: Typical framework for transformer-based SR methods, where the block-level and layer-level improvements made by our HiT-SR are colored red. SA, FFN, and LN indicate self-attention, feed-forward network, and layer normalization, respectively."
+ ],
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+ "text": "GMSA remains quadratic complexity and suffers from information trade-off due to channel splitting. In our approach, a spatial-channel correlation (SCC) method is designed to efficiently utilize hierarchical features with linear complexity to window sizes, while preserving spatial and channel information.",
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+ "text": "We first introduce the basic methodology of HiT-SR in Sec. 3.1 and then present the block-level and layer-level designs in HiT-SR in Sec. 3.2 and 3.3, respectively.",
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+ "text": "We first review the commonly adopted pipeline of transformer-based SR methods [10, 26, 55]. As depicted in Fig. 2, popular transformer-based SR framework often consists of convolutional layers to extract shallow features $F_{S} \\in \\mathbb{R}^{C \\times H \\times W}$ from LR input images $I_{LR} \\in \\mathbb{R}^{3 \\times H \\times W}$ , a feature extraction module to aggregate deep image features $F_{D} \\in \\mathbb{R}^{C \\times H \\times W}$ via transformer blocks (TBs), and a reconstruction module to restore HR images $I_{HR} \\in \\mathbb{R}^{3 \\times sH \\times sW}$ ( $s$ denotes upscaling factor) from shallow and deep features. In the feature extraction module, TBs are generally built with cascaded transformer layers (TLs) followed by convolutional layers, where each TL consists of self-attention (SA), feed-forward network (FFN), and layer normalization (LN). Since the computational complexity of SA is quadratic to input sizes [14], window partition is often employed in TL to limit SA on local regions, which is known as window self-attention (W-SA) [26, 30]. Although W-SA alleviates the computational burden, its receptive field is restricted to small local regions, hindering SR networks from utilizing long-range dependencies and multi-scale information.",
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+ "text": "To efficiently aggregate hierarchical features, we propose a general strategy to convert the above SR framework to hierarchical transformers. As shown in Fig. 2, our approach mainly contains improvements in two aspects: (i) Instead of using",
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+ "Fig. 4: Layer-Level design in HiT-SR composed of dual feature extraction (DFE), spatial and channel self-correlation (S-SC and C-SC). DFE is designed to extract features from spatial and channel domains. S-SC and C-SC are proposed to efficiently aggregate hierarchical information with linear computational complexity to window sizes."
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+ "text": "fixed small window sizes for all TLs, we apply hierarchical windows to different TLs in the block level, enabling HiT-SR to establish long-range dependency and gather multi-scale information; (ii) To overcome the computational burden caused by large windows, we replace the W-SA in TLs with a novel spatial-channel correlation (SCC) method, which better supports window scaling with linear computational complexity to window sizes. Based on the above strategies, HiT-SR not only gains better performance by exploiting hierarchical features but also maintains computational efficiency thanks to SCC.",
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+ "type": "text",
+ "text": "3.2 Block-Level Design: Hierarchical Windows",
+ "text_level": 1,
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+ "text": "At the block level, we assign hierarchical windows to different TLs, collecting multi-scale features. Given a base window size $h_B \\times w_B$ , we set the window size $h_i \\times w_i$ for the $i$ -th TL to",
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+ "text": "\n$$\nh _ {i} = \\alpha_ {i} h _ {B}, \\quad w _ {i} = \\alpha_ {i} w _ {B}, \\tag {1}\n$$\n",
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+ "text": "where $\\alpha_{i} > 0$ is the hierarchical ratio for the $i$ -th TL.",
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+ "text": "Expanding Windows. To better aggregate hierarchical features, we arrange the windows with an expanding strategy. As illustrated in Fig. 3, we first use small window sizes in the initial layers to gather the most relevant features from local regions, and",
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+ {
+ "type": "image",
+ "img_path": "images/25e4515216c1375348156d0e76e23376160115283b8930f8ea0e5e30dfce8237.jpg",
+ "image_caption": [
+ "Fig.3: Block-Level design in HiT-SR. Hierarchical windows are applied to different transformer layers (TLs) to aggregate features with expanding receptive fields."
+ ],
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+ "type": "header",
+ "text": "HiT-SR: Hierarchical Transformer for Efficient Image Super-Resolution",
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+ "type": "page_number",
+ "text": "5",
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+ "type": "text",
+ "text": "then gradually expand the window size to utilize the information gained from long-range dependencies. Previous approaches often apply shifting and masking operations on fixed small windows [9, 10, 26, 55] to enlarge receptive fields, but these operations are time-consuming and inefficient in practice. Compared with them, our approach directly utilizes the cascaded TLs to form a hierarchical feature extractor, enabling small to large receptive fields while maintaining overall efficiency. Fig. 1 shows the better performance of our HiT-SR methods with $\\sim 7\\times$ faster speeds over the original models.",
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+ "type": "text",
+ "text": "3.3 Layer-Level Design: Spatial-Channel Correlation",
+ "text_level": 1,
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+ "type": "text",
+ "text": "At the layer level, we propose spatial-channel correlation (SCC) to efficiently leverage spatial and temporal information from hierarchical inputs. As depicted in Fig. 4, our SCC mainly consists of dual feature extraction (DFE), spatial self-correlation (S-SC), and channel self-correlation (C-SC). Besides, unlike commonly adopted multi-head strategies [9,26,39], different correlation head strategies are applied to S-SC and C-SC to better utilize image features.",
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+ "type": "text",
+ "text": "Dual Feature Extraction. Linear layers are often employed for feature projection, which only extracts channel information and neglects to model spatial relations. Instead, we propose DFE with a two-branch design to utilize features from two domains. As shown in Fig. 4, DFE consists of a convolutional branch to exploit spatial information and a linear branch to extract channel features.",
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+ "type": "text",
+ "text": "Given an input feature $X \\in \\mathbb{R}^{C \\times H \\times W}$ , the output of DFE is computed as",
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+ "type": "equation",
+ "text": "\n$$\n\\begin{array}{l} \\operatorname {D F E} (X) = X _ {c h} \\odot X _ {s p}, \\quad \\text {w i t h} \\\\ X _ {c h} = \\operatorname {L i n e a r} (X), X _ {s p} = \\operatorname {C o n v} (X), \\tag {2} \\\\ \\end{array}\n$$\n",
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+ "type": "text",
+ "text": "where $\\odot$ denotes element-wise multiplication. The reshaped channel feature $X_{ch} \\in \\mathbb{R}^{HW \\times C}$ and spatial feature $X_{sp} \\in \\mathbb{R}^{HW \\times C}$ are captured by linear and convolutional layers, respectively. In the spatial branch, we use an hourglass structure to stack three convolutional layers with the hidden dimension reduced by ratio $r$ for efficiency. Finally, the spatial and channel features interact with each other by multiplication, yielding the DFE output.",
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+ {
+ "type": "text",
+ "text": "Unlike standard SA methods that predict queries, keys, and values by linear projection, we equate keys with values as they both reflect the intrinsic properties of input features, and only estimate queries $Q \\in \\mathbb{R}^{HW \\times \\frac{C}{2}}$ and values $V \\in \\mathbb{R}^{HW \\times \\frac{C}{2}}$ by splitting the DFE output as displayed in Fig. 4,",
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+ {
+ "type": "equation",
+ "text": "\n$$\n[ Q, V ] = \\operatorname {D F E} (X), \\tag {3}\n$$\n",
+ "text_format": "latex",
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+ "type": "text",
+ "text": "which reduces the information redundancy caused by key estimation. Then, we partition queries and keys to non-overlapped windows according to the assigned window size, e.g., $Q_{i}$ , $V_{i} \\in \\mathbb{R}^{h_{i}w_{i} \\times \\frac{C}{2}}$ for the $i$ -th TL (the number of windows is omitted for simplicity), and use the partitioned queries and values for the subsequent self-correlation.",
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+ "type": "text",
+ "text": "Spatial Self-Correlation. Compared with W-SA, our S-SC aggregates spatial information in an efficient manner. Considering the expanding window sizes in",
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+ {
+ "type": "page_number",
+ "text": "6",
+ "bbox": [
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+ "page_idx": 5
+ },
+ {
+ "type": "header",
+ "text": "Xiang Zhang, Yulun Zhang, and Fisher Yu",
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+ "page_idx": 5
+ },
+ {
+ "type": "table",
+ "img_path": "images/965bddd0b61d009e070be82b59becf405eb77aa6a25acdb6719ddbf668ae2ef4.jpg",
+ "table_caption": [
+ "Table 1: Complexity of different layer types. $C, N, h, w$ correspond to channel dimension, window number, height, and width. $m = \\max (C, h_{\\downarrow} w_{\\downarrow})$ is upper bounded."
+ ],
+ "table_footnote": [],
+ "table_body": "