SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} \left\| \underline {{\mathbf {W}}} ^ {(l)} - \underline {{\mathbf {W}}} _ {r _ {l}} ^ {(l)} \right\| _ {\mathrm {F}} = \left\| \underline {{\mathbf {W}}} ^ {(l)} \times_ {3} \mathbf {M} - \underline {{\mathbf {W}}} _ {r _ {l}} ^ {(l)} \times_ {3} \mathbf {M} \right\| _ {\mathrm {F}} \\ \leq \sqrt {\sum_ {k = 1} ^ {c} \sum_ {j = r _ {l} + 1} ^ {\min  \left\{d _ {l} , d _ {l - 1} \right\}} \sigma_ {j} ^ {2} \left(\underline {{\mathbf {W}}} ^ {(l)} \times_ {3} \mathbf {M}\right) _ {, ; , k}} \\ \leq \sqrt {\sum_ {k = 1} ^ {c} \sum_ {j = r _ {l} + 1} ^ {\min  \left\{d _ {l} , d _ {l - 1} \right\}} \left(V _ {0} \cdot j ^ {- \alpha}\right) ^ {2}} \\ \leq \sqrt {\sum_ {k = 1} ^ {c} \frac {1}{2 \alpha - 1} V _ {0} ^ {2} (r _ {l} - 1) ^ {1 - 2 \alpha}} \\ \stackrel {(i)} {\leq} \sqrt {\frac {\mathsf {c}}{2 \alpha - 1}} V _ {0} r _ {l} ^ {(1 - 2 \alpha) / 2} := \delta_ {l} ^ {\mathrm {F}}, \\ \end{array}
$$
where  $(i)$  holds because of Lemma 41.
We also have a spectral norm bound for  $\underline{\mathbf{W}}^{(l)} - \underline{\mathbf{W}}_{r_l}^{(l)}$  according to [20, Theorem 3.7] under Assumption 13 as follows
$$
\left\| \underline {{\mathbf {W}}} ^ {(l)} - \underline {{\mathbf {W}}} _ {r _ {l}} ^ {(l)} \right\| _ {\mathrm {s p}} \leq V _ {0} (r _ {l} + 1) ^ {- \alpha} := \delta_ {l} ^ {\mathrm {s p}},
$$
$$
\sqrt {c} \delta_ {l} ^ {\mathrm {s p}} \leq \delta_ {l} ^ {\mathrm {F}}. \tag {61}
$$
Consider function  $g(\underline{\mathbf{x}}) = g(\underline{\mathbf{x}}; \underline{\mathbf{W}}_{\mathbf{r}})$  parameterized by  $\underline{\mathbf{W}}_{\mathbf{r}} = (\underline{\mathbf{W}}_{r_1}^{(1)}, \dots, \underline{\mathbf{W}}_{r_L}^{(L)}, \mathbf{w})$  as the function whose t-product layer weights are low-tubal-rank approximations of  $f(\underline{\mathbf{x}}; \underline{\mathbf{W}})$ . Let  $\tilde{f}(\underline{\mathbf{x}}, y) = \inf_{R_{\mathrm{a}}(\underline{\mathbf{x}} - \underline{\mathbf{x}}') \leq \xi} yf(\underline{\mathbf{x}}')$  and  $\tilde{g}(\underline{\mathbf{x}}, y) = \inf_{R_{\mathrm{a}}(\underline{\mathbf{x}} - \underline{\mathbf{x}}') \leq \xi} yg(\underline{\mathbf{x}}')$  denote the adversarial versions of  $f$  and  $g$ , respectively.
This helps us bounding  $\hat{\tau}$  as follows
Let  $h_l(\underline{\mathbf{x}}^\xi) = \mathbf{w}^\top_{\mathrm{vec}}\left(\sigma (\underline{\mathbf{W}}_{r_l}^{(L)})_*M\sigma (\underline{\mathbf{W}}_{r - 1}^{(L - 1)}*)_M\dots *_M\sigma (\underline{\mathbf{W}}_r^{(l + 1)})_*M\sigma (\underline{\mathbf{W}}^ {(l)}*)_M\dots *_M\sigma (\underline{\mathbf{W}}^{(1)}*)_M\right)$ .
$$
\begin{array}{l} \left\| \sigma (\underline {{\mathbf {W}}} _ {r _ {l}} ^ {(l)} * _ {M} \sigma (\underline {{\mathbf {W}}} ^ {(l - 1)} * _ {M} \dots * _ {M} \sigma (\underline {{\mathbf {W}}} ^ {(1 - 1)} * _ {M} \underline {{\mathbf {x}}} _ {i} ^ {\xi}) \cdot \cdot \cdot)) - \sigma (\underline {{\mathbf {W}}} ^ {(l)} * _ {M} \sigma (\underline {{\mathbf {W}}} ^ {(l - 1)} * _ {M} \dots * _ {M} \sigma (\underline {{\mathbf {W}}} ^ {(1)} * _ {M} \underline {{\mathbf {x}}} _ {i} ^ {\xi}) \cdot \cdot \cdot)) \right\| _ {\mathrm {F}} \\ \leq \left\| \underline {{\mathbf {W}}} _ {r _ {l}} ^ {(l)} * _ {M} \sigma (\underline {{\mathbf {W}}} ^ {(l - 1)} * _ {M} \dots * _ {M} \sigma (\underline {{\mathbf {W}}} ^ {(1)} * _ {M} \underline {{\mathbf {x}}} _ {i} ^ {\xi}) \dots) - \underline {{\mathbf {W}}} ^ {(l)} * _ {M} \sigma (\underline {{\mathbf {W}}} ^ {(l - 1)} * _ {M} \dots * _ {M} \sigma (\underline {{\mathbf {W}}} ^ {(1)} * _ {M} \underline {{\mathbf {x}}} _ {i} ^ {\xi}) \dots)\right\| _ {\mathrm {F}} \\ = \left\| \left(\underline {{\mathbf {W}}} _ {r _ {l}} ^ {(l)} - \underline {{\mathbf {W}}} ^ {(l)}\right) * _ {M} \sigma \left(\underline {{\mathbf {W}}} ^ {(l - 1)} * _ {M} \dots * _ {M} \sigma \left(\underline {{\mathbf {W}}} ^ {(1)} * _ {M} \underline {{\mathbf {x}}} _ {i} ^ {\xi}\right) \dots\right) \right\| _ {\mathrm {F}} \\ \stackrel {(i)} {\leq} \left\| \underline {{\mathbf {W}}} _ {r _ {l}} ^ {(l)} - \underline {{\mathbf {W}}} ^ {(l)} \right\| _ {\mathrm {s p}} \left\| \sigma \left(\underline {{\mathbf {W}}} ^ {(l - 1)} * _ {M} \dots * _ {M} \sigma \left(\underline {{\mathbf {W}}} ^ {(1)} * _ {M} \underline {{\mathbf {x}}} _ {i} ^ {\xi}\right) \dots\right) \right\| _ {\mathrm {F}} \\ \leq \delta_ {l} ^ {\mathrm {s p}} \prod_ {l ^ {\prime} = 1} ^ {l} \left\| \underline {{\mathbf {W}}} ^ {(l ^ {\prime})} \right\| _ {\mathrm {F}} B _ {x, R _ {\mathrm {a}}, \xi}, \\ \end{array}
$$
where in (i) we relax the inequality by using the tensor spectral norm of instead of the F-norm of  $\underline{\mathbf{W}}_{r_l}^{(l)} - \underline{\mathbf{W}}^{(l)}$  which leads to a smaller upper bound. Thus, we have
$$
\left| h _ {l} \left(\underline {{\mathbf {x}}} _ {i} ^ {\xi}\right) - h _ {l - 1} \left(\underline {{\mathbf {x}}} _ {i} ^ {\xi}\right) \right| \leq B _ {\mathbf {w}} \prod_ {j \neq l} B _ {j} \delta_ {l} ^ {\mathrm {s p}} B _ {x, R _ {\mathrm {a}}, \xi} = \frac {\delta_ {l} ^ {\mathrm {s p}} B _ {\tilde {f}}}{B _ {l}}.
$$
$$
| f (\underline {{\mathbf {x}}} _ {i} ^ {\xi}) - g (\underline {{\mathbf {x}}} _ {i} ^ {\xi}) | \leq \sum_ {l = 1} ^ {L} | h _ {l} (\underline {{\mathbf {x}}} _ {i} ^ {\xi}) - h _ {l - 1} (\underline {{\mathbf {x}}} _ {i} ^ {\xi}) | \leq \sum_ {l = 1} ^ {L} \frac {\delta_ {l} ^ {\mathrm {s p}} B _ {\tilde {f}}}{B _ {l}}.
$$
$$
\hat {\mathfrak {r}} = \sum_ {l = 1} ^ {L} \frac {\delta_ {l} ^ {\mathrm {s p}} B _ {\tilde {f}}}{B _ {l}} = V _ {0} B _ {\tilde {f}} \sum_ {l = 1} ^ {L} \frac {(r _ {l} + 1) ^ {- \alpha}}{B _ {l}}. \tag {62}
$$
Then we can construct an  $\epsilon$ -cover of  $\mathfrak{F}_{\delta, \mathbf{r}}^{\mathrm{adv}}$  composed of  $\tilde{g}(\underline{\mathbf{x}}; \underline{\mathbf{W}}_{\mathbf{r}})$  by carefully setting the value of rank parameter  $\mathbf{r} = (r_1, \dots, r_L)^\top$  according to the covering accuracy  $\epsilon$ . Directly setting  $\frac{\delta_l^{\mathrm{sp}} D_{\tilde{f}}}{2B_l} = \epsilon / L$ , we obtain
$$
r _ {l} = \min  \left\{\left\lceil A _ {l} \cdot \epsilon^ {- \frac {1}{\alpha}} \right\rceil - 1, d _ {l}, d _ {l - 1} \right\} \leq \min  \left\{A _ {l} \cdot \epsilon^ {- \frac {1}{\alpha}}, d _ {l}, d _ {l - 1} \right\},
$$
$$
A _ {l} = \left(\frac {B _ {l}}{L V _ {0} B _ {\tilde {f}}}\right) ^ {- \frac {1}{\alpha}}.
$$
Then, we obtain the covering entropy of  $\mathfrak{F}_{\delta ,\mathbf{r}}^{\mathrm{adv}}$  by
$$
\begin{array}{l} \log \mathsf {N} (\mathfrak {F} _ {\delta , \mathbf {r}} ^ {\mathrm {a d v}}, \| \cdot \| _ {S}, \epsilon) \leq \log \prod_ {l = 1} ^ {L} \left(\frac {9 B _ {l}}{\delta_ {l} ^ {\mathrm {F}}}\right) ^ {r _ {l} (d _ {l} + d _ {l - 1} + 1) \mathsf {c}} \\ \stackrel {(i)} {\leq} \log \prod_ {l = 1} ^ {L} \left(\frac {9 B _ {l}}{\sqrt {\mathsf {C}} \delta_ {l} ^ {\mathrm {s p}}}\right) ^ {r _ {l} (d _ {l} + d _ {l - 1} + 1) \mathsf {c}} \\ \leq \sum_ {l = 1} ^ {L} r _ {l} \left(d _ {l} + d _ {l - 1} + 1\right) \mathrm {c} \log \left(9 L B _ {\tilde {f}} / (\sqrt {\mathrm {c}} \epsilon)\right) \\ \leq \sum_ {l} \left(A _ {l} \epsilon^ {- \frac {1}{\alpha}}\right) \left(d _ {l} + d _ {l - 1} + 1\right) \mathsf {c} \left(\log \left(\epsilon^ {- 1}\right) + \log \left(9 L B _ {\tilde {f}} / (\sqrt {\mathsf {c}})\right)\right), \\ \end{array}
$$
where  $(ii)$  holds by Eq. (61).
Since  $\mathfrak{F}_{\mathbf{r}}^{\mathrm{adv}}\subset \mathfrak{F}_{\delta ,\mathbf{r}}^{\mathrm{adv}}$  we have
$$
\log \mathsf {N} (\mathfrak {F} _ {\mathbf {r}} ^ {\mathrm {a d v}}, \| \cdot \| _ {S}, \epsilon) \leq \log \mathsf {N} (\mathfrak {F} _ {\delta , \mathbf {r}} ^ {\mathrm {a d v}}, \| \cdot \| _ {S}, \epsilon) \leq \sum_ {l = 1} ^ {L} r _ {l} (d _ {l} + d _ {l - 1} + 1) \mathsf {c} \log (9 L B _ {\bar {f}} / (\sqrt {\mathsf {c}} \epsilon))
$$
To use Lemma 27, we bound  $\log \mathsf{N}(\mathfrak{F}_{\delta ,\mathbf{r}}^{\mathrm{adv}},\| \cdot \| _S,\epsilon /2) + \mathsf{N}(\mathfrak{F}_{\mathbf{r}}^{\mathrm{adv}},\| \cdot \| _S,\epsilon /2)$  when  $\epsilon \geq 2 / N$  as follows:
$$
\log \mathsf {N} (\mathfrak {F} _ {\delta , \mathbf {r}} ^ {\mathrm {a d v}}, \| \cdot \| _ {S}, \epsilon / (2)) + \mathsf {N} (\mathfrak {F} _ {\mathbf {r}} ^ {\mathrm {a d v}}, \| \cdot \| _ {S}, \epsilon / 2) \leq a _ {1} + a _ {2} \log (\epsilon^ {- 1}) + a _ {3} \epsilon^ {- \frac {1}{\alpha}},
$$
$$
a _ {1} = \log (9 L B _ {\tilde {f}} / \sqrt {\mathsf {c}}) a _ {2},
$$
$$
a _ {2} = \mathrm {c} \sum_ {l = 1} ^ {L} r _ {l} \left(d _ {l} + d _ {l - 1} + 1\right),
$$
$$
a _ {3} = \left(\log N + \log \left(9 L B _ {\tilde {f}} / \sqrt {\mathrm {c}}\right)\right) \mathrm {c} \sum_ {l = 1} ^ {L} A _ {l} \left(d _ {l} + d _ {l - 1} + 1\right).
$$
For simplicity, further let
$$
E _ {1} = \frac {a _ {1} + a _ {2} \log N}{N} = \frac {\mathsf {c} \sum_ {l = 1} ^ {L} r _ {l} \left(d _ {l} + d _ {l - 1} + 1\right)}{N} \log \left(9 N L B _ {\tilde {f}} / \sqrt {\mathsf {c}}\right), \tag {64}
$$
$$
E _ {2} = \frac {a _ {3}}{N} = \frac {\mathsf {c} \sum_ {l = 1} ^ {L} A _ {l} (d _ {l} + d _ {l - 1} + 1)}{N} \log (9 N L B _ {\tilde {f}} / \sqrt {\mathsf {c}}).
$$
Then, according to Lemma 27, we have
$$
\mathfrak {r} _ {*} ^ {2} (t) \leq C \left\{B _ {\tilde {f}} E _ {1} + \frac {1 + t B _ {\tilde {f}}}{N}, \quad E _ {2} ^ {\frac {2 \alpha}{2 \alpha + 1}} \left(B _ {\tilde {f}} ^ {\frac {2 \alpha - 1}{2 \alpha + 1}} + 1\right) \right\}, \tag {65}
$$
$$
\begin{array}{l} \Phi (\mathfrak {r}) + L _ {\ell} \dot {\mathfrak {r}} \sqrt {\frac {t}{N}} + \frac {t L _ {\ell} B _ {\tilde {f}}}{N} \\ \leq 2 L _ {\ell} \phi (\mathfrak {r}) + 2 L _ {\ell} (\mathfrak {r} + \mathfrak {r} _ {*}) \sqrt {\frac {t}{N}} + \frac {t L _ {\ell} B _ {\tilde {f}}}{N} \tag {66} \\ \leq C _ {q} L _ {\ell} \max \left\{\frac {1}{N} + B _ {\tilde {f}} E _ {1} + (\mathfrak {r} + \mathfrak {r} _ {*}) \sqrt {E _ {1}}, E _ {2} ^ {\frac {2 \alpha}{2 \alpha + 1}} B _ {\tilde {f}} ^ {\frac {2 \alpha - 1}{2 \alpha + 1}} + (\mathfrak {r} + \mathfrak {r} _ {*}) ^ {\frac {2 \alpha}{2 \alpha + 1}} \sqrt {E _ {2}} \right\} \\ + 2 L _ {\ell} (\hat {\mathbf {r}} + \mathbf {r} _ {*}) \sqrt {\frac {t}{N}} + \frac {t L _ {\ell} B _ {\tilde {f}}}{N}. \\ \end{array}
$$
$$
(\hat {\mathfrak {r}} + \mathfrak {r} _ {*}) \sqrt {E _ {1}} = \hat {\mathfrak {r}} \sqrt {E _ {1}} + \mathfrak {r} _ {*} \sqrt {E _ {1}} \leq \hat {\mathfrak {r}} \sqrt {E _ {1}} + \frac {1}{2} \mathfrak {r} _ {*} ^ {2} + \frac {1}{2} E _ {1},
$$
$$
(\hat {\mathfrak {r}} + \mathfrak {r} _ {*}) ^ {\frac {2 \alpha}{2 \alpha + 1}} \sqrt {E _ {2}} \leq \hat {\mathfrak {r}} ^ {\frac {2 \alpha}{2 \alpha + 1}} \sqrt {E _ {2}} + \mathfrak {r} _ {*} ^ {\frac {2 \alpha}{2 \alpha + 1}} \sqrt {E _ {2}},
$$
$$
\left(\hat {\mathfrak {r}} + \mathfrak {r} _ {*}\right) \sqrt {\frac {t}{N}} \leq \hat {\mathfrak {r}} \sqrt {\frac {t}{N}} + \frac {1}{2} \left(\mathfrak {r} _ {*} ^ {2} + \frac {t}{N}\right).
$$
Then by simple calculation, we have
$$
\begin{array}{l} \Phi (\mathfrak {r}) + L _ {\ell} \dot {\mathfrak {r}} \sqrt {\frac {t}{N}} + \frac {t L _ {\ell} B _ {\tilde {f}}}{N} \\ \stackrel {(i)} {\leq} C _ {\alpha} L _ {\ell} \Bigg \{B _ {\tilde {f}} E _ {1} + \hat {\mathfrak {r}} \sqrt {E _ {1}} + E _ {2} ^ {\frac {2 \alpha}{2 \alpha + 1}} \left(B _ {\tilde {f}} ^ {\frac {2 \alpha - 1}{2 \alpha + 1}} + 1\right) + \hat {\mathfrak {r}} ^ {\frac {2 \alpha}{2 \alpha + 1}} \sqrt {E _ {2}} + \hat {\mathfrak {r}} \sqrt {\frac {t}{N}} + \frac {1 + t B _ {\tilde {f}}}{N} \Bigg \}. \\ \end{array}
$$
Proof of Corollary 15. The bound in Corollary 15 can be directly obtained if we choose the parameter  $\mathbf{r}$  of tubal ranks in  $\mathfrak{F}_{\mathbf{r}}$  by  $r_l = \min \{\lceil (LV_0B_{\tilde{f}}B_l^{-1})^{1 / \alpha}\rceil ,d_l,d_{l - 1}\}$ .
F Useful Notions and Lemmas
In this section, we provide several notions and lemmas which are used in the previous analysis.
F.1 Tools for Analyzing General DNNs
We briefly list the tools used in this paper for analyzing the generalization error of general DNNs, including Rademacher complexity, covering number, and concentration inequalities, etc..
Definition 13 (Rademacher complexity). Given an i.i.d. sample  $S \coloneqq \{(\underline{\mathbf{x}}_i, y_i)\}_{i=1}^N$  of size  $N$  and a function class  $\mathcal{H}$ , the empirical Rademacher complexity of  $\mathcal{H}$  is defined as
$$
\hat {R} _ {S} (\mathcal {H}) := \mathbb {E} _ {\varepsilon_ {1}, \dots , \varepsilon_ {N}} \left[ \sup  _ {h \in \mathcal {H}} \frac {1}{N} \varepsilon_ {i} h (\underline {{\mathbf {x}}} _ {i}, y _ {i}) \right],
$$
where  $\varepsilon_1, \dots, \varepsilon_N$  are i.i.d. Rademacher variables, i.e.,  $\varepsilon_i$  equals to 1 or -1 with equal probability. The average Rademacher complexity is further defined as
$$
\bar {R} _ {N} = \mathbb {E} _ {S} \hat {R} _ {S} (\mathcal {H}).
$$
Lemma 29 ([4]). Given an i.i.d. sample  $S \coloneqq \{(\underline{\mathbf{x}}_i, y_i)\}_{i=1}^N$  of size  $N$ , a loss function  $\ell(h(\cdot), y)$  taking values in  $[0, B]$ , the generalization error of any function  $f$  in hypothesis set  $\mathcal{F}$  satisfies
$$
\mathcal {L} (f) \leq \hat {\mathcal {L}} (f) + 2 \hat {R} _ {S} (\ell \circ \mathcal {F}) + 3 B \sqrt {\frac {t}{2 N}}, \tag {67}
$$
with probability at least  $1 - e^{-t}$  for all  $t \geq 0$ .
Lemma 30 (Talagrand's contraction lemma [44]). Given function set  $\mathcal{F}$  and  $L_{\ell}$ -Lipschitz function  $\ell$ , for a function sets defined as  $l_{\mathcal{F}} \coloneqq \{\ell \circ f \mid f \in \mathcal{F}\}$ , we have
$$
\hat {R} _ {S} (\mathcal {F}) \leq L _ {\ell} \hat {R} _ {S} (\mathcal {F}).
$$
Definition 14 (ε-covering net). Let  $\epsilon >0$  and  $(\mathcal{X},d(\cdot ,\cdot))$  be a metric space, where  $d(\cdot ,\cdot)$  is a (pseudo)-metric. We say  $\mathcal{Z}\subset \mathcal{X}$  is an  $\epsilon$ -covering net of  $\mathcal{X}$ , if for any  $x\in \mathcal{X}$ , there exists  $z\in \mathcal{Z}$  such that  $d(x,z)\leq \epsilon$ . Define the smallest  $|\mathcal{Z}|$  as the  $\epsilon$ -covering number of  $\mathcal{X}$  and denote as  $\mathsf{N}(\mathcal{X},d(\cdot ,\cdot),\epsilon)$ .
Given a training dataset  $S = \{\underline{\mathbf{x}}_i, y_i\}_{i=1}^N$  and a function set  $\mathcal{F}$ . Consider the output space of the space of  $\mathcal{F}$  restricted on  $S$ , i.e.,  $\mathcal{F}|_S = \left\{\left(f(\underline{\mathbf{x}}_1, y_1), \dots, f(\underline{\mathbf{x}}_N, y_N)\right)^\top \mid f \in \mathcal{F}\right\}$ . Then, define a pseudo-norm of  $\mathcal{F}|_S$  as:  $\|f\|_S := N^{-1} \sqrt{\sum_{i=1}^N f(\underline{\mathbf{x}}_i, y_i)^2}$ . Then, the Rademacher complexity of  $\mathcal{F}$  could be upper bounded by the  $\epsilon$ -covering number of  $\mathcal{F}$  under the empirical  $l_2$ -pseudo-metric by the Dudley's inequality as follows:
Lemma 31 (Dudley's integral inequality [47]). The Rademacher complexity  $\hat{R}_S(\mathcal{F})$  satisfies
$$
\hat {R} _ {S} (\mathcal {F}) \leq \inf  _ {\delta > 0} \left[ 8 \delta + \frac {1 2}{\sqrt {N}} \int_ {\delta} ^ {\max  _ {f \in \mathcal {F}} \| f \| _ {S}} \sqrt {\log \mathrm {N} (\mathcal {F} , \| \cdot \| _ {S} , \epsilon)} d \epsilon \right]. \tag {68}
$$
Lemma 32 (Covering number of norm balls [47]). Let  $\mathbf{B}$  be a  $l_p$ -norm ball with radius  $W$ . Let  $d(\mathbf{x}_1, \mathbf{x}_2) = \|\mathbf{x}_1 - \mathbf{x}_2\|_p$ . Define the  $\epsilon$ -covering number of  $\mathbf{B}$  as  $\mathsf{N}(\mathbf{B}, d(\cdot, \cdot), \epsilon)$ , we have
$$
\mathsf {N} (\mathbf {B}, d (\cdot , \cdot), \epsilon) \leq \left(1 + \frac {2 W}{\epsilon}\right) ^ {d}. \tag {69}
$$
Lemma 33 (Covering number of low-tubal-rank tensors). For the set of tensors  $\mathbf{T}_r \coloneqq \{\mathbf{T} \in \mathbb{R}^{m \times n \times c} \mid r_t(\mathbf{T}) \leq r, \| \mathbf{T}\|_{\mathrm{F}} \leq 1\}$  with  $r \leq \min \{m, n\}$ , its  $\epsilon$ -covering number can be upper bounded by
$$
\mathsf {N} \left(\mathbf {T} _ {r}, \| \cdot \| _ {\mathbf {F}}, \epsilon\right) \leq \left(\frac {9}{\epsilon}\right) ^ {(m + n + 1) r c}. \tag {70}
$$
Proof. Consider the reduced t-SVD [20] of a tensor  $\underline{\mathbf{X}} = \underline{\mathbf{U}} *_{M}\underline{\mathbf{S}} *_{M}\underline{\mathbf{V}}^{\top} \in \mathbf{T}_{r}$ , where  $^{*}_{M}$  denotes the t-product induced by the linear transform  $M(\cdot)$  defined in Eq. (1),  $\underline{\mathbf{U}} \in \mathbb{R}^{m \times r \times c}$  and  $\underline{\mathbf{V}} \in \mathbb{R}^{n \times r \times c}$  are (semi)-t-orthogonal tensors, and  $\underline{\mathbf{S}} \in \mathbb{R}^{r \times r \times c}$  is an f-diagonal tensor. As  $\underline{\mathbf{T}} \in \overline{\mathbf{T}}_r$ , we have  $\| \underline{\mathbf{S}} \|_{\mathrm{F}} = \| \underline{\mathbf{X}} \|_{\mathrm{F}} \leq 1$ . The idea is to cover  $\mathbf{T}_r$  by covering the set of factor tensors  $\underline{\mathbf{U}}, \underline{\mathbf{V}}$  and  $\underline{\mathbf{S}}$ .
Let  $\mathbf{D} \subset \mathbb{R}^{r \times r \times c}$  be the set of f-diagonal tensors with F-norm equal to one. We take  $\mathbf{D}^c$  to be an  $\epsilon/3$ -covering net for  $\mathbf{D}$ . Then by Lemma 32, we have  $|\mathbf{D}^c| \leq (9/\epsilon)^{rc}$ . Next, let  $\mathbf{O}_{m,r} = \{\underline{\mathbf{U}} \in \mathbb{R}^{m \times r \times c} | \underline{\mathbf{U}} *_M \underline{\mathbf{U}}^\top = \underline{\mathbf{I}}\}$ . To cover  $\mathbf{O}_{m,r}$ , we consider the  $\| \cdot \|_{\infty,2,2}$ -norm defined as
$$
\| \underline {{\mathbf {X}}} \| _ {\infty , 2, 2} := \max _ {i} \left\| \underline {{\mathbf {X}}} _ {:, i,:} \right\| _ {\mathrm {F}}.
$$
Let  $\mathbf{Q}_{m,r} \coloneqq \{\underline{\mathbf{X}} \in \mathbb{R}^{m \times r \times c} \mid \| \underline{\mathbf{X}} \|_{\infty,2,2} \leq 1\}$ . Then, we have  $\mathbf{O}_{m,r} \subset \mathbf{Q}_{m,r}$  by the definition of t-orthogonal tensors. Letting  $\mathbf{Q}_{m,r}^c$  be an  $\epsilon/3$ -covering net of  $\mathbf{Q}_{m,r}$ , then we obtain  $|\mathbf{Q}_{m,r}^c| \leq (9/\epsilon)^{mr_c}$  by Theorem 32. Similarly, an  $\epsilon/3$ -covering net of  $\mathbf{Q}_{n,r}$  satisfies  $|\mathbf{Q}_{n,r}^c| \leq (9/\epsilon)^{nrc}$ .
Now construct a set  $\mathbf{T}_r^c = \{\underline{\mathbf{U}}^c *_M\underline{\mathbf{S}}^c *_M(\underline{\mathbf{V}}^c)^\top \mid \underline{\mathbf{U}}^c\in \mathbf{Q}_{m,r}^c,\underline{\mathbf{S}}^c\in \mathbf{D}^c,\underline{\mathbf{V}}^c\in \mathbf{Q}_{n,r}^c\}$ . Then, we have
$$
\left| \mathbf {T} _ {r} ^ {c} \right| = \left| \mathbf {Q} _ {m, r} ^ {c} \right| \cdot \left| \underline {{\mathbf {S}}} ^ {c} \right| \cdot \left| \mathbf {Q} _ {n, r} ^ {c} \right| \leq (9 / \epsilon) ^ {(m + n + 1) r c}.
$$
Net, we will show that  $\mathbf{T}_r^c$  is an  $\epsilon$ -covering net of  $\mathbf{T}_r$ , i.e., for any  $\underline{\mathbf{X}} \in \mathbf{T}_r$ , there is an  $\underline{\mathbf{X}}^c \in \mathbf{T}_r^c$  satisfying  $\| \underline{\mathbf{X}} - \underline{\mathbf{X}}^c \|_{\mathrm{F}} \leq \epsilon$ .
Given  $\underline{\mathbf{X}}\in \mathbf{T}_r$ , consider its reduced t-SVD as  $\underline{\mathbf{X}} = \underline{\mathbf{U}} *_{M}\underline{\mathbf{S}} *_{M}\underline{\mathbf{V}}^{\top}\in \mathbf{T}_{r}$ . Then, there exists  $\underline{\mathbf{X}}^{c} = \underline{\mathbf{U}}^{c}*_{M}\underline{\mathbf{S}}^{c}*_{M}(\underline{\mathbf{V}}^{c})^{\top}$  with  $\underline{\mathbf{U}}^{c}\in \mathbf{Q}_{m,r}^{c}$ ,  $\underline{\mathbf{S}}^{c}\in \mathbf{D}^{c},\underline{\mathbf{V}}^{c}\in \mathbf{Q}_{n,r}^{c}$  satisfying  $\| \underline{\mathbf{U}} -\underline{\mathbf{U}}^{c}\|_{\infty ,2,2}\leq \epsilon /3$ ,  $\| \underline{\mathbf{S}} -\underline{\mathbf{S}}^{c}\|_{F}\leq \epsilon /3$ , and  $\| \underline{\mathbf{V}} -\underline{\mathbf{V}}^{c}\|_{\infty ,2,2}\leq \epsilon /3$ . This gives
For the first term, note that since  $\mathbf{V}$  is a t-orthogonal tensor,
$$
\left\| \left(\underline {{\mathbf {U}}} - \underline {{\mathbf {U}}} ^ {c}\right) * _ {M} \underline {{\mathbf {S}}} * _ {M} \underline {{\mathbf {V}}} ^ {\top} \right\| _ {\mathbf {F}} = \| \left(\underline {{\mathbf {U}}} - \underline {{\mathbf {U}}} ^ {c}\right) * _ {M} \underline {{\mathbf {S}}} \| _ {\mathbf {F}}.
$$
$$
\begin{array}{l} \left\| \left(\underline {{\mathbf {U}}} - \underline {{\mathbf {U}}} ^ {c}\right) * _ {M} \underline {{\mathbf {S}}} \right\| _ {\mathrm {F}} ^ {2} = \sum_ {i = 1} ^ {r} \left\| \left(\underline {{\mathbf {U}}} - \underline {{\mathbf {U}}} ^ {c}\right) _ {:, i,:} * _ {M} \underline {{\mathbf {S}}} _ {i, i,:} \right\| _ {\mathrm {F}} ^ {2} \\ = \sum_ {i = 1} ^ {r} \| (\underline {{\mathbf {U}}} - \underline {{\mathbf {U}}} ^ {c}) _ {:, i,:} \| _ {\mathrm {F}} ^ {2} \| \underline {{\mathbf {S}}} _ {i, i,:} \| _ {\mathrm {F}} ^ {2} \\ \leq \left(\sum_ {i = 1} ^ {r} \left\| \underline {{\mathbf {S}}} _ {i, i,:} \right\| _ {\mathrm {F}} ^ {2}\right) \max  _ {i} \| (\underline {{\mathbf {U}}} - \underline {{\mathbf {U}}} ^ {c}) _ {:, i,:} \| _ {\mathrm {F}} ^ {2} \\ \leq \left\| \underline {{\mathbf {S}}} \right\| _ {\mathrm {F}} ^ {2} \left\| \underline {{\mathbf {U}}} - \underline {{\mathbf {U}}} ^ {c} \right\| _ {\infty , 2, 2} \\ \leq (\epsilon / 3) ^ {2}. \\ \end{array}
$$
Hence,  $\left\| (\underline{\mathbf{U}} - \underline{\mathbf{U}}^c) *_M \underline{\mathbf{S}} *_M \underline{\mathbf{V}}^\top \right\|_{\mathrm{F}} \leq \epsilon / 3$ . Similarly, we have  $\left\| \underline{\mathbf{U}}^c *_M \underline{\mathbf{S}}^c *_M (\underline{\mathbf{V}} - \underline{\mathbf{V}}^c)^\top \right\|_{\mathrm{F}} \leq \epsilon / 3$ . The middle term can be bounded  $\left\| \underline{\mathbf{U}}^c *_M (\underline{\mathbf{S}} - \underline{\mathbf{S}}^c) *_M \underline{\mathbf{V}}^\top \right\|_{\mathrm{F}} \leq \| \underline{\mathbf{S}} - \underline{\mathbf{S}}^c \|_{\mathrm{F}} \leq \epsilon / 3$  due to the property of t-orthogonal tensors  $\underline{\mathbf{U}}^c$  and  $\underline{\mathbf{V}}^c$  [20]. Therefore, for any  $\underline{\mathbf{X}} \in \mathbf{T}_r$ , there is an  $\underline{\mathbf{X}}^c \in \mathbf{T}_r^c$  satisfying  $\| \underline{\mathbf{X}} - \underline{\mathbf{X}}^c \|_{\mathrm{F}} \leq \epsilon$ .
Lemma 34 (Covering number bounds for composition and addition [10]). Let  $\mathcal{F}_1$  and  $\mathcal{F}_2$  be classes of functions on normed space  $(\mathcal{X},\| \cdot \|_{\mathcal{X}})\to (\mathcal{Y},\| \cdot \|_{\mathcal{Y}})$  and let  $\mathcal{F}$  be a class of c-Lipschitz functions  $(\mathcal{V},\| \cdot \|_{\mathcal{V}})\to (\mathcal{Z},\| \cdot \|_{\mathcal{Z}})$ . Then for any  $X\in \mathcal{X}^N$  and  $\epsilon_{\mathcal{F}_1},\epsilon_{\mathcal{F}_2},\epsilon_{\mathcal{F}} > 0$ , it holds that
$$
\mathsf {N} \left(\left\{f _ {1} + f _ {2} \mid f _ {1} \in \mathcal {F} _ {1}, f _ {2} \in \mathcal {F} _ {2} \right\}, \epsilon_ {\mathcal {F} _ {1}} + \epsilon_ {\mathcal {F} _ {2}}, \| \cdot \| _ {X}\right) \leq \mathsf {N} \left(\mathcal {F} _ {1}, \epsilon_ {\mathcal {F} _ {1}}, \| \cdot \| _ {X}\right) \mathsf {N} \left(\mathcal {F} _ {2}, \epsilon_ {\mathcal {F} _ {2}}, \| \cdot \| _ {X}\right).
$$
$$
\mathsf {N} (\{f \circ f _ {1} \mid f \in \mathcal {F}, f _ {1} \in \mathcal {F} _ {1} \}, \epsilon_ {\mathcal {F}} + c \epsilon_ {\mathcal {F} _ {1}}, \| \cdot \| _ {X}) \leq \mathsf {N} (\mathcal {F} _ {1}, \epsilon_ {\mathcal {F} _ {1}}, \| \cdot \| _ {X}) \sup  _ {f _ {1} \in \mathcal {F} _ {1}} \mathsf {N} (\mathcal {F}, \epsilon_ {\mathcal {F}}, \| \cdot \| _ {f _ {1} (X)}).
$$
Specifically, if  $\mathcal{F} = \{f\}$  is a singleton, we have
$$
\mathsf {N} \left(\left\{f \circ f _ {1} \mid f \in \mathcal {F}, f _ {1} \in \mathcal {F} _ {1} \right\}, c \epsilon_ {\mathcal {F} _ {1}}, \| \cdot \| _ {X}\right) \leq \mathsf {N} \left(\mathcal {F} _ {1}, \epsilon_ {\mathcal {F} _ {1}}, \| \cdot \| _ {X}\right).
$$