SlowGuess/NeurIPS_2023_OCR huggingface dataset:
where the second and the third inequalities follow from the facts that  $1 - q \geq \exp(-2q)$  for all  $0 \leq q \leq \frac{1}{2}$  and  $\exp(-q) \geq 1 - q$  for all  $0 \leq q \leq 1$ , respectively. It is straightforward to observe that when  $m \geq \log_{c_0}\left(\frac{2K}{\delta}\right)$ , we have  $\frac{2K}{c_0^m} \leq \delta$ . With  $\min_{k \in [K]} \left|\mathcal{X}_k^f\right| = m_0$ , taking up to  $m \leq m_0$  unlabeled samples for each  $k \in [K]$ , we can achieve  $\delta$  as small as  $\delta_{\mathrm{min}} = \frac{2K}{c_0^{m_0}}$ .
C Proofs and Discussions for Section 4.1
In Appendix C.1, we proceed with the proof of low clustering error guarantee (Theorem 4.1) for minimizing the population RKD loss (Equation (1)). In Appendix C.2, we extend the discussion in Remark 4.1 on the limitation of spectral clustering alone in the end-to-end setting via an illustrative toy counter-example. In Appendix C.3, we clarify the connection between our results and corresponding spectral graph theoretic notions of sparsest  $k$ -partitions.
Given any labeling function  $y: \mathcal{X} \to [K]$ , let  $\overrightarrow{y}: \mathcal{X} \to \{0,1\}^K$  be the one-hot encoded labeling function such that for  $y(\mathcal{X}) \in [K]^{|\mathcal{X}|}$ ,  $\overrightarrow{y}(\mathcal{X}) \in \{0,1\}^{|\mathcal{X}| \times K}$ . Further, given  $f \in \mathcal{F}$ , we denote  $\mathbf{U}_{\mathcal{X}} \triangleq \mathbf{D}(\mathcal{X})^{1/2} \overrightarrow{\overrightarrow{y}_f}(\mathcal{X})$ ,  $\mathbf{F}_{\mathcal{X}} \triangleq \mathbf{D}(\mathcal{X})^{1/2} f(\mathcal{X})$ , and  $\mathbf{Y}_{\mathcal{X}} \triangleq \mathbf{D}(\mathcal{X})^{1/2} \overrightarrow{y_*}(\mathcal{X})$  such that  $\mathbf{U}_{\mathcal{X}}, \mathbf{F}_{\mathcal{X}}, \mathbf{Y}_{\mathcal{X}} \in \mathbb{R}^{|\mathcal{X}| \times K}$ .
Since  $\mathbf{L}(\mathcal{X}) = \mathbf{I} - \overline{\mathbf{W}}(\mathcal{X})$ , the classical argument with Gershgorin circle theorem [Golub and Van Loan, 2013, Theorem 7.2.1] implies that  $0 \preccurlyeq \mathbf{L}(\mathcal{X}) \preccurlyeq 2\mathbf{I}$ , whereas  $k_{\psi}$  being positive semi-definite further implies that  $\overline{\mathbf{W}}(\mathcal{X}) \succcurlyeq 0$  and therefore  $0 \preccurlyeq \mathbf{L}(\mathcal{X}) \preccurlyeq \mathbf{I}$  (see the proof of Claim C.3). Consider the spectral decomposition of  $\mathbf{L}(\mathcal{X}) = \mathbf{I} - \overline{\mathbf{W}}(\mathcal{X})$ ,
$$
\mathbf {L} (\mathcal {X}) = \sum_ {i = 1} ^ {| \mathcal {X} |} \lambda_ {i} \cdot \mathbf {v} _ {i} \mathbf {v} _ {i} ^ {\top} \quad \text {s . t .} \quad \mathbf {D} (\mathcal {X}) ^ {- 1 / 2} \mathbf {W} (\mathcal {X}) \mathbf {D} (\mathcal {X}) ^ {- 1 / 2} = \sum_ {i = 1} ^ {| \mathcal {X} |} (1 - \lambda_ {i}) \cdot \mathbf {v} _ {i} \mathbf {v} _ {i} ^ {\top}, \tag {13}
$$
where  $\{\mathbf{v}_i\in \mathbb{R}^{|\mathcal{X}|}\}_{i\in [|\mathcal{X}|]}$  are the orthonormal eigenvectors associated with  $0 = \lambda_{1}\leq \dots \leq \lambda_{|\mathcal{X}|}\leq 1$ , respectively, that form a basis of  $\mathbb{R}^{|X|}$ .
Lemma C.1. Given any  $f\in \mathcal{F}$  that satisfies Assumption 4.2, we have
$$
P \left(M (f)\right) = \mathbb {E} _ {\mathbf {x} \sim P} \left[ \mathbb {1} \left\{\widetilde {y} _ {f} (\mathbf {x}) \neq y _ {*} (\mathbf {x}) \right\} \right] \leq 2 \max  \left(\frac {\beta^ {2}}{\gamma^ {2}}, 1\right) \cdot \min  _ {\mathbf {Z} \in \mathbb {R} ^ {K \times K}} \| \mathbf {Y} _ {\mathcal {X}} - \mathbf {F} _ {\mathcal {X}} \mathbf {Z} \| _ {F} ^ {2}.
$$
Proof of Lemma C.1. Without ambiguity, we overload the notation of each sample  $\mathbf{x} \in \mathcal{X}$  as the corresponding index in  $[|\mathcal{X}|]$  such that, given any sample subset  $S = \{\mathbf{s}_i\}_{i \in [|S|]} \subset \mathcal{X}$ , following the MATLAB notation for matrix indexing,  $\Pi_S \triangleq \mathbf{I}_{|\mathcal{X}|}(:, S) \in \mathbb{R}^{|\mathcal{X}| \times |S|}$  where  $\mathbf{I}_{|\mathcal{X}|}$  denotes the  $|\mathcal{X}| \times |\mathcal{X}|$  identity matrix. With respect to  $S$ , we also denote  $\mathbf{F}_S \triangleq \mathbf{F}_{\mathcal{X}}(S,:) = \Pi_S^\top \mathbf{F}_{\mathcal{X}}$  and  $\mathbf{Y}_S \triangleq \mathbf{Y}_{\mathcal{X}}(S,:) = \Pi_S^\top \mathbf{Y}_{\mathcal{X}}$ .
We first show that for any  $\mathbf{S} = [\mathbf{s}_1; \ldots; \mathbf{s}_K] \in \mathcal{X}^K$  that satisfies  $\mathrm{rank}(\mathbf{F}_S) = K$ ,
$$
\left\| \mathbf {Y} _ {\mathcal {X}} - \mathbf {F} _ {\mathcal {X}} \mathbf {F} _ {S} ^ {- 1} \mathbf {Y} _ {S} \right\| _ {F} ^ {2} \leq 2 \min  _ {\mathbf {Z} \in \mathbb {R} ^ {K \times K}} \| \mathbf {Y} _ {\mathcal {X}} - \mathbf {F} _ {\mathcal {X}} \mathbf {Z} \| _ {F} ^ {2}. \tag {14}
$$
To see this, we start by replacing  $\mathbf{Z}$  on the right-hand side of Equation (14) with the corresponding least square solution such that
$$
\min _ {\mathbf {Z} \in \mathbb {R} ^ {K} \times K} \| \mathbf {Y} _ {\mathcal {X}} - \mathbf {F} _ {\mathcal {X}} \mathbf {Z} \| _ {F} ^ {2} = \left\| \mathbf {Y} _ {\mathcal {X}} - \mathbf {F} _ {\mathcal {X}} \mathbf {F} _ {\mathcal {X}} ^ {\dagger} \mathbf {Y} _ {\mathcal {X}} \right\| _ {F} ^ {2}
$$
where  $\mathbf{F}_{\mathcal{X}}^{\dagger} = \left(\mathbf{F}_{\mathcal{X}}^{\top}\mathbf{F}_{\mathcal{X}}\right)^{-1}\mathbf{F}_{\mathcal{X}}^{\top}$ . Then, we can decompose the left-hand side of Equation (14) via the orthogonal projection  $\mathbf{F}_{\mathcal{X}}\mathbf{F}_{\mathcal{X}}^{\dagger}$ :
$$
\begin{array}{l} \left\| \mathbf {Y} _ {\mathcal {X}} - \mathbf {F} _ {\mathcal {X}} \mathbf {F} _ {S} ^ {- 1} \mathbf {Y} _ {S} \right\| _ {F} ^ {2} = \left\| \left(\mathbf {I} - \mathbf {F} _ {\mathcal {X}} \mathbf {F} _ {\mathcal {X}} ^ {\dagger}\right) \left(\mathbf {Y} _ {\mathcal {X}} - \mathbf {F} _ {\mathcal {X}} \mathbf {F} _ {S} ^ {- 1} \mathbf {Y} _ {S}\right) \right\| _ {F} ^ {2} + \left\| \mathbf {F} _ {\mathcal {X}} \mathbf {F} _ {\mathcal {X}} ^ {\dagger} \left(\mathbf {Y} _ {\mathcal {X}} - \mathbf {F} _ {\mathcal {X}} \mathbf {F} _ {S} ^ {- 1} \mathbf {Y} _ {S}\right) \right\| _ {F} ^ {2} \\ = \left\| \mathbf {Y} _ {\mathcal {X}} - \mathbf {F} _ {\mathcal {X}} \mathbf {F} _ {\mathcal {X}} ^ {\dagger} \mathbf {Y} _ {\mathcal {X}} \right\| _ {F} ^ {2} + \| (\mathbf {P} _ {F} - \mathbf {P} _ {S}) \mathbf {Y} _ {\mathcal {X}} \| _ {F} ^ {2}, \\ \end{array}
$$
where  $\mathbf{P}_F\triangleq \mathbf{F}_{\mathcal{X}}\mathbf{F}_{\mathcal{X}}^\dagger = \mathbf{F}_{\mathcal{X}}\left(\mathbf{F}_{\mathcal{X}}^\top \mathbf{F}_{\mathcal{X}}\right)^{-1}\mathbf{F}_{\mathcal{X}}^\top$  and  $\mathbf{P}_S\triangleq \mathbf{F}_{\mathcal{X}}\left(\Pi_S^\top \mathbf{F}_{\mathcal{X}}\right)^{-1}\Pi_S^\top$ . By observing that
$$
\mathbf {P} _ {S} \mathbf {P} _ {F} = \mathbf {F} _ {\mathcal {X}} \left(\boldsymbol {\Pi} _ {S} ^ {\top} \mathbf {F} _ {\mathcal {X}}\right) ^ {- 1} \boldsymbol {\Pi} _ {S} ^ {\top} \mathbf {F} _ {\mathcal {X}} \left(\mathbf {F} _ {\mathcal {X}} ^ {\top} \mathbf {F} _ {\mathcal {X}}\right) ^ {- 1} \mathbf {F} _ {\mathcal {X}} ^ {\top} = \mathbf {P} _ {F} = \mathbf {P} _ {F} \mathbf {P} _ {F},
$$
we can upper bound  $\| (\mathbf{P}_F - \mathbf{P}_S)\mathbf{Y}_{\mathcal{X}}\| _F^2$  such that
$$
\begin{array}{l} \left\| \left(\mathbf {P} _ {F} - \mathbf {P} _ {S}\right) \mathbf {Y} _ {\mathcal {X}} \right\| _ {F} ^ {2} = \left\| \left(\mathbf {P} _ {F} - \mathbf {P} _ {S}\right) \left(\mathbf {I} - \mathbf {P} _ {F}\right) \mathbf {Y} _ {\mathcal {X}} \right\| _ {F} ^ {2} \\ \leq \left\| \left(\mathbf {I} - \mathbf {P} _ {F}\right) \mathbf {Y} _ {\chi} \right\| _ {F} ^ {2} \\ = \left\| \mathbf {Y} _ {\mathcal {X}} - \mathbf {F} _ {\mathcal {X}} \mathbf {F} _ {\mathcal {X}} ^ {\dagger} \mathbf {Y} _ {\mathcal {X}} \right\| _ {F} ^ {2}, \\ \end{array}
$$
which leads to Equation (14).
Now, we consider the sample subset  $\mathbf{S} \in \mathcal{X}^K$  described in Assumption 4.2. For given  $f \in \mathcal{F}$ , partitioning  $\mathcal{X}$  into the majority and minority subsets,  $\mathcal{X} \backslash M(f)$  and  $M(f)$ , respectively yields
$$
\begin{array}{l} \left\| \mathbf {Y} _ {\mathcal {X}} - \mathbf {F} _ {\mathcal {X}} \mathbf {F} _ {S} ^ {- 1} \mathbf {Y} _ {S} \right\| _ {F} ^ {2} \\ = (1 - P (M (f))) \cdot \mathbb {E} _ {\mathbf {x} \sim P (\mathbf {x})} \left[ \left\| \overrightarrow {y _ {*}} (\mathbf {x}) - \overrightarrow {y _ {*}} (\mathbf {S}) ^ {\top} f (\mathbf {S}) ^ {- \top} f (\mathbf {x}) \right\| _ {2} ^ {2} \mid \mathbf {x} \notin M (f) \right] \\ + P (M (f)) \cdot \mathbb {E} _ {\mathbf {x} \sim P (\mathbf {x})} \left[ \left\| \overrightarrow {y _ {*}} (\mathbf {x}) - \overrightarrow {y _ {*}} (\mathbf {S}) ^ {\top} f (\mathbf {S}) ^ {- \top} f (\mathbf {x}) \right\| _ {2} ^ {2} \mid \mathbf {x} \in M (f) \right] \\ \geq P (M (f)) \cdot \mathbb {E} _ {\mathbf {x} \sim P (\mathbf {x})} \left[ \left| \left| \overrightarrow {y _ {*} ^ {\prime}} (\mathbf {x}) - \overrightarrow {y _ {*} ^ {\prime}} (\mathbf {S}) ^ {\top} f (\mathbf {S}) ^ {- \top} f (\mathbf {x}) \right| \right| _ {2} ^ {2} \right\rvert   \mathbf {x} \in M (f) \Bigg ]. \\ \end{array}
$$
Since  $\mathbf{s}_k \notin M(f)$ , we have  $\overrightarrow{y_*(S)} = \overrightarrow{\widetilde{y}_f}(\mathbf{S})$  whose rows lie in the canonical bases of  $\{\mathbf{e}_k \in \mathbb{R}^K\}_{k \in [K]}$ ; whereas for  $\mathbf{x} \in M(f)$ , we have  $y_*(\mathbf{x}) \neq \widetilde{y}_f(\mathbf{x})$ . Therefore, in the case when  $y_*(\mathbf{x}) = y_*(\mathbf{s}_k)$  for some  $k \in [K]$ ,
$$
\begin{array}{l} \left\| \overrightarrow {y _ {*}} (\mathbf {x}) - \overrightarrow {y _ {*}} (\mathbf {S}) ^ {\top} f (\mathbf {S}) ^ {- \top} f (\mathbf {x}) \right\| _ {2} ^ {2} \\ = \left\| f (\mathbf {S}) ^ {- \top} \left(f (\mathbf {S}) ^ {\top} \overrightarrow {y _ {*}} (\mathbf {S}) \overrightarrow {y _ {*}} (\mathbf {x}) - f (\mathbf {x})\right) \right\| _ {2} ^ {2} \\ \geq \sigma_ {K} \left(f (\mathbf {S}) ^ {- \top}\right) ^ {2} \cdot \left\| f (\mathbf {x}) - \sum_ {k \in [ K ]: y _ {*} (\mathbf {x}) = y _ {*} (\mathbf {s} _ {k})} f (\mathbf {s} _ {k}) \right\| _ {2} ^ {2}, \\ \end{array}
$$
where  $\sigma_K\left(f(\mathbf{S})^{-\top}\right) = 1 / \beta$ , and with  $\widetilde{y}_f(\mathbf{s}_k) = y_*(\mathbf{s}_k) = y_*(\mathbf{x}) \neq \widetilde{y}_f(\mathbf{x})$  implying  $y_{f}(\mathbf{s}_{k}) \neq y_{f}(\mathbf{x})$ ,
$$
\left\| f (\mathbf {x}) - \sum_ {k \in [ K ]: y _ {*} (\mathbf {x}) = y _ {*} (\mathbf {s} _ {k})} f (\mathbf {s} _ {k}) \right\| _ {2} ^ {2} \geq \gamma^ {2},
$$
$$
\left\| \overrightarrow {y _ {*}} (\mathbf {x}) - \overrightarrow {y _ {*}} (\mathbf {S}) ^ {\top} f (\mathbf {S}) ^ {- \top} f (\mathbf {x}) \right\| _ {2} ^ {2} \geq \frac {\gamma^ {2}}{\beta^ {2}}.
$$
Meanwhile, in the case when  $y_{*}(\mathbf{x}) \neq y_{*}(\mathbf{s}_{k})$  for all  $k \in [K]$ , we have  $\overrightarrow{y_{*}}(\mathbf{S})\overrightarrow{y_{*}}(\mathbf{x}) = \mathbf{0}$ , and therefore
$$
\left\| \overrightarrow {y _ {*}} (\mathbf {x}) - \overrightarrow {y _ {*}} (\mathbf {S}) ^ {\top} f (\mathbf {S}) ^ {- \top} f (\mathbf {x}) \right\| _ {2} ^ {2} \geq \left\| \overrightarrow {y _ {*}} (\mathbf {x}) ^ {\top} \left(\overrightarrow {y _ {*}} (\mathbf {x}) - \overrightarrow {y _ {*}} (\mathbf {S}) ^ {\top} f (\mathbf {S}) ^ {- \top} f (\mathbf {x})\right) \right\| _ {2} ^ {2} = 1.
$$
Overall, we've shown that
$$
\mathbb {E} _ {\mathbf {x} \sim P (\mathbf {x})} \left[ \left\| \overrightarrow {y _ {*}} (\mathbf {x}) - \overrightarrow {y _ {*}} (\mathbf {S}) ^ {\top} f (\mathbf {S}) ^ {- \top} f (\mathbf {x}) \right\| _ {2} ^ {2} \mid \mathbf {x} \in M (f) \right] \geq \min  \left(\frac {\gamma^ {2}}{\beta^ {2}}, 1\right),
$$
which leads to
$$
P (M (f)) \leq \max  \left(\frac {\beta^ {2}}{\gamma^ {2}}, 1\right) \left\| \mathbf {Y} _ {\mathcal {X}} - \mathbf {F} _ {\mathcal {X}} \mathbf {F} _ {S} ^ {- 1} \mathbf {Y} _ {S} \right\| _ {F} ^ {2}. \tag {15}
$$
Finally, combining Equation (14) and Equation (15) completes the proof.
Lemma C.2. With  $\mathbf{y}_k\in [0,1]^{\mathcal{X}}$  denoting the  $k$  -th column of  $\mathbf{Y}_{\mathcal{X}}\triangleq \mathbf{D}(\mathcal{X})^{1 / 2}\overrightarrow{y_{*}} (\mathcal{X})\in [0,1]^{|\mathcal{X}|\times K},$
$$
\sum_ {k \in [ K ]} \mathbf {y} _ {k} ^ {\top} \mathbf {L} (\mathcal {X}) \mathbf {y} _ {k} = \alpha .
$$
Proof of Lemma C.2. It is sufficient to observe that, by construction, for every  $k \in [K]$ ,
$$
\begin{array}{l} \mathbf {y} _ {k} ^ {\top} \mathbf {L} (\mathcal {X}) \mathbf {y} _ {k} = \mathbf {y} _ {k} ^ {\top} \mathbf {y} _ {k} - \mathbf {y} _ {k} ^ {\top} \mathbf {D} (\mathcal {X}) ^ {- 1 / 2} \mathbf {W} (\mathcal {X}) \mathbf {D} (\mathcal {X}) ^ {- 1 / 2} \mathbf {y} _ {k} \\ = \sum_ {\mathbf {x} \in \mathcal {X}} w _ {\mathbf {x}} \cdot \mathbb {1} \left\{y _ {*} (\mathbf {x}) = k \right\} - \sum_ {\mathbf {x} \in \mathcal {X}} \sum_ {\mathbf {x} ^ {\prime} \in \mathcal {X}} w _ {\mathbf {x x} ^ {\prime}} \cdot \mathbb {1} \left\{y _ {*} (\mathbf {x}) = k \right\} \cdot \mathbb {1} \left\{y _ {*} \left(\mathbf {x} ^ {\prime}\right) = k \right\} \\ = \sum_ {\mathbf {x} \in \mathcal {X}} \mathbb {1} \left\{y _ {*} (\mathbf {x}) = k \right\} \left(w _ {\mathbf {x}} - \sum_ {\mathbf {x} ^ {\prime} \in \mathcal {X}} w _ {\mathbf {x x} ^ {\prime}} \cdot \mathbb {1} \left\{y _ {*} (\mathbf {x}) = y _ {*} (\mathbf {x} ^ {\prime}) \right\}\right) \\ = \sum_ {\mathbf {x} \in \mathcal {X}} \mathbb {1} \left\{y _ {*} (\mathbf {x}) = k \right\} \sum_ {\mathbf {x} ^ {\prime} \in \mathcal {X}} w _ {\mathbf {x x} ^ {\prime}} \cdot \mathbb {1} \left\{y _ {*} (\mathbf {x}) \neq y _ {*} \left(\mathbf {x} ^ {\prime}\right) \right\}. \\ \end{array}
$$
Meanwhile, since  $\sum_{\mathbf{x} \in \mathcal{X}} \sum_{\mathbf{x}' \in \mathcal{X}} w_{\mathbf{xx}'} = 1$ , we have  $\alpha = \frac{1}{2} \sum_{k \in [K]} \sum_{\mathbf{x} \in \mathcal{X}_k} \sum_{\mathbf{x}' \notin \mathcal{X}_k} w_{\mathbf{xx}'}$  and
$$
\begin{array}{l} \sum_ {k \in [ K ]} \mathbf {y} _ {k} ^ {\top} \mathbf {L} (\mathcal {X}) \mathbf {y} _ {k} = \sum_ {\mathbf {x} \in \mathcal {X}} \sum_ {\mathbf {x} ^ {\prime} \in \mathcal {X}} w _ {\mathbf {x x} ^ {\prime}} \cdot \mathbb {1} \left\{y _ {*} (\mathbf {x}) \neq y _ {*} \left(\mathbf {x} ^ {\prime}\right) \right\} \sum_ {k \in [ K ]} \mathbb {1} \left\{y _ {*} (\mathbf {x}) = k \right\} \\ = \sum_ {\mathbf {x} \in \mathcal {X}} \sum_ {\mathbf {x} ^ {\prime} \in \mathcal {X}} w _ {\mathbf {x x} ^ {\prime}} \cdot \mathbb {1} \left\{y _ {*} (\mathbf {x}) \neq y _ {*} \left(\mathbf {x} ^ {\prime}\right) \right\} \\ = \frac {1}{2} \sum_ {k \in [ K ]} \sum_ {\mathbf {x} \in \mathcal {X} _ {k}} \sum_ {\mathbf {x} ^ {\prime} \notin \mathcal {X} _ {k}} w _ {\mathbf {x x} ^ {\prime}} = \alpha . \\ \end{array}
$$
Claim C.3. For all  $f \in \mathcal{F}_{\mathbf{L}(\mathcal{X})}$  minimizing Equation (1), Range  $\left(\mathbf{D}(\mathcal{X})^{\frac{1}{2}}f(\mathcal{X})\right) =$  span  $\{\mathbf{v}_1, \ldots, \mathbf{v}_K\}$ .
Proof of Claim C.3. Since  $0 \precsim \mathbf{L}(\mathcal{X}) \precsim 2\mathbf{I}$ , we have  $-\mathbf{I} \precsim \overline{\mathbf{W}}(\mathcal{X}) \precsim \mathbf{I}$ . Meanwhile, let  $k_{\psi}(\mathcal{X}, \mathcal{X}) \in \mathbb{R}^{|\mathcal{X}| \times |\mathcal{X}|}$  be the population kernel matrix such that  $k_{\psi}(\mathcal{X}, \mathcal{X})_{\mathbf{x}\mathbf{x}'} = k_{\psi}(\mathbf{x}, \mathbf{x}')$ . Then by definition,  $k_{\psi}(\mathcal{X}, \mathcal{X}) = \mathbf{D}(\mathcal{X})^{-1}\mathbf{W}(\mathcal{X})\mathbf{D}(\mathcal{X})^{-1}$ , and therefore
$$
\mathbf {I} \succcurlyeq \overline {{\mathbf {W}}} (\mathcal {X}) = \mathbf {D} (\mathcal {X}) ^ {- \frac {1}{2}} \mathbf {W} (\mathcal {X}) \mathbf {D} (\mathcal {X}) ^ {- \frac {1}{2}} = \mathbf {D} (\mathcal {X}) ^ {\frac {1}{2}} k _ {\psi} (\mathcal {X}, \mathcal {X}) \mathbf {D} (\mathcal {X}) ^ {\frac {1}{2}} \succcurlyeq 0.
$$
That is, the eigenvalues of  $\mathbf{L}(\mathcal{X})$  and those of  $\overline{\mathbf{W}} (\mathcal{X})$  satisfies
$$
\lambda_ {i} \triangleq \lambda_ {i} (\mathbf {L} (\mathcal {X})) = 1 - \lambda_ {| \mathcal {X} | - i + 1} (\overline {{\mathbf {W}}} (\mathcal {X})) \quad \forall i \in [ | \mathcal {X} | ],
$$
while the eigenvectors are shared such that, with the spectral decomposition of  $\mathbf{L}(\mathcal{X})$  in Equation (13),
$$
\overline {{\mathbf {W}}} (\mathcal {X}) = \sum_ {i = 1} ^ {| \mathcal {X} |} (1 - \lambda_ {i}) \cdot \mathbf {v} _ {i} \mathbf {v} _ {i} ^ {\top}.
$$
Therefore, the  $K$  eigenvalues of  $\overline{\mathbf{W}} (\mathcal{X})$  of maximal modulus correspond exactly to the  $K$  eigenvalues of  $\mathbf{L}(\mathcal{X})$  that are closest to 0, the eigenvalues whose eigenspaces are relevant for spectral clustering.
Recalling  $\mathbf{F}_{\mathcal{X}} \triangleq \mathbf{D}(\mathcal{X})^{\frac{1}{2}} f(\mathcal{X})$ , we can rewrite Equation (1) as
$$
R (f) = \left\| \overline {{\mathbf {W}}} (\mathcal {X}) - \mathbf {F} _ {\mathcal {X}} \mathbf {F} _ {\mathcal {X}} ^ {\top} \right\| _ {F} ^ {2}.
$$
It follows directly from the Eckart-Young-Mirsky theorem [Eckart and Young, 1936] that  $\mathrm{Range}(\mathcal{F}_{\mathcal{X}})$  is the subspace spanned by the eigenvectors  $\mathbf{V}_K = [\mathbf{v}_1,\dots ,\mathbf{v}_K]\in \mathbb{R}^{|\mathcal{X}|\times K}$  associated with the maximum  $K$  eigenvalues of  $\overline{\mathbf{W}} (\mathcal{X})$ :
$$
\mathcal {F} _ {\mathbf {L} (\mathcal {X})} = \left\{f \in \mathcal {F} \middle |   \exists \mathbf {Z} \in \mathbb {R} ^ {K \times K} \text {s . t .} \mathbf {D} (\mathcal {X}) ^ {\frac {1}{2}} f (\mathcal {X}) = \mathbf {V} _ {K} \mathbf {Z} \right\}.
$$
Proof of Theorem 4.1. Recall  $\mathbf{F}_{\mathcal{X}} \triangleq \mathbf{D}(\mathcal{X})^{1/2} f(\mathcal{X})$ ,  $\mathbf{Y}_{\mathcal{X}} \triangleq \mathbf{D}(\mathcal{X})^{1/2} \overrightarrow{y_*(\mathcal{X})}$ , and let  $\mathbf{y}_k \in [0,1]^{|\mathcal{X}|}$  be the  $k$ -th column of  $\mathbf{Y}_{\mathcal{X}}$ . Leveraging Lemma C.1, we have for any  $f \in \mathcal{F}$ ,
$$
\begin{array}{l} P \left(M (f)\right) \leq 2 \max  \left(\frac {\beta^ {2}}{\gamma^ {2}}, 1\right) \min  _ {\mathbf {Z} \in \mathbb {R} ^ {K \times K}} \| \mathbf {Y} _ {\mathcal {X}} - \mathbf {F} _ {\mathcal {X}} \mathbf {Z} \| _ {F} ^ {2} \\ = 2 \max \left(\frac {\beta^ {2}}{\gamma^ {2}}, 1\right) \min _ {\left\{\mathbf {z} _ {k} \in \mathbb {R} ^ {K} \right\} _ {k \in [ K ]}} \sum_ {k \in [ K ]} \| \mathbf {y} _ {k} - \mathbf {F} _ {\mathcal {X}} \mathbf {z} _ {k} \| _ {2} ^ {2}. \\ \end{array}
$$
Recall that the orthonormal eigenvectors  $\{\mathbf{v}_i\in \mathbb{R}^{|\mathcal{X}|}\}_{i\in [|X|]}$  associated with eigenvalues  $0 = \lambda_{1}\leq \dots \leq \lambda_{|\mathcal{X}|}$  of  $\mathbf{L}(\mathcal{X})$  form a basis of  $\mathbb{R}^{|X|}$ . For every  $k\in [K]$ ,  $(\mathbf{y}_k - \mathbf{F}_X\mathbf{z}_k)\in \mathbb{R}^{|\mathcal{X}|}$  can be decomposed as the linear combination of  $\{\mathbf{v}_i\}_{i\in [|X|]}$  such that
$$
\left\| \mathbf {y} _ {k} - \mathbf {F} _ {\mathcal {X}} \mathbf {z} _ {k} \right\| _ {2} ^ {2} = \sum_ {i = 1} ^ {| \mathcal {X} |} \left(\mathbf {v} _ {i} ^ {\top} \left(\mathbf {y} _ {k} - \mathbf {F} _ {\mathcal {X}} \mathbf {z} _ {k}\right)\right) ^ {2}
$$
Meanwhile, Claim C.3 implies that, for  $f \in \mathcal{F}_{\mathbf{L}(\mathcal{X})}$ , with  $\mathbf{V}_K = [\mathbf{v}_1, \ldots, \mathbf{v}_K] \in \mathbb{R}^{|\mathcal{X}| \times K}$ , we have  $\mathrm{Range}(\mathbf{F}_{\mathcal{X}}) = \mathrm{Range}(\mathbf{V}_K)$ . That is,  $\mathbf{v}_i^\top \mathbf{F}_{\mathcal{X}} \mathbf{z}_k = 0$  for all  $i > K$ , and by taking
$$
\mathbf {z} _ {k} = \left(\mathbf {V} _ {K} ^ {\top} \mathbf {F} _ {\mathcal {X}}\right) ^ {- 1} \mathbf {V} _ {K} ^ {\top} \mathbf {y} _ {k} \quad \Rightarrow \quad \sum_ {i = 1} ^ {K} \left(\mathbf {v} _ {i} ^ {\top} \left(\mathbf {y} _ {k} - \mathbf {F} _ {\mathcal {X}} \mathbf {z} _ {k}\right)\right) ^ {2} = 0.
$$
Overall, for every  $k \in [K]$ , there exists  $\mathbf{z}_k \in \mathbb{R}^K$  such that
$$
\left\| \mathbf {y} _ {k} - \mathbf {F} _ {\mathcal {X}} \mathbf {z} _ {k} \right\| _ {2} ^ {2} = \sum_ {i = K + 1} ^ {| \mathcal {X} |} \left(\mathbf {v} _ {i} ^ {\top} \mathbf {y} _ {k}\right) ^ {2} \leq \frac {1}{\lambda_ {K + 1}} \sum_ {i = K + 1} ^ {| \mathcal {X} |} \lambda_ {i} \cdot \left(\mathbf {v} _ {i} ^ {\top} \mathbf {y} _ {k}\right) ^ {2} \leq \frac {\mathbf {y} _ {k} ^ {\top} \mathbf {L} (\mathcal {X}) \mathbf {y} _ {k}}{\lambda_ {K + 1}},
$$
and Lemma C.2 implies that
$$
\min  _ {\left\{\mathbf {z} _ {k} \in \mathbb {R} ^ {K} \right\} _ {k \in [ K ]}} \sum_ {k \in [ K ]} \| \mathbf {y} _ {k} - \mathbf {F} _ {\mathcal {X}} \mathbf {z} _ {k} \| _ {2} ^ {2} \leq \sum_ {k \in [ K ]} \frac {\mathbf {y} _ {k} ^ {\top} \mathbf {L} (\mathcal {X}) \mathbf {y} _ {k}}{\lambda_ {K + 1}} = \frac {\alpha}{\lambda_ {K + 1}},
$$
C.2 Limitation of Spectral Clustering Alone
To ground the discussion in Remark 4.1, here, we raise a toy counter-example where spectral clustering alone fails to satisfy Assumption 4.2 and suffers from a large clustering error. Intuitively, such a pitfall of spectral clustering is caused by a suboptimal choice of basis for the predictions  $f(\mathcal{X})$  in Equation (1) due to the inherited rotation invariance of the minimizers of  $R(f)$ .
Example C.1 (Pitfall of spectral clustering alone). We consider a binary classification problem with two balanced ground truth classes  $\{\mathcal{X}_1,\mathcal{X}_2\}$  perfectly partitioned in  $G_{\mathcal{X}}$  as disconnected components. Then, when the prediction  $y_{f}(\mathbf{x})\triangleq \mathrm{argmax}_{k\in [K]}f(\mathbf{x})_{k}$  is made with a uniformly random break of ties, Equation (1) alone suffers from  $\mu \left(\mathcal{F}_{\mathbf{L}(\mathcal{X})}\right)\geq \frac{1}{4}$  in expectation, due to the violation of Assumption 4.2.
Nevertheless, with as few as one labeled sample per class  $\{(\mathbf{x}_i,i)\mid i = 1,2\}$ , weak supervision via the cross-entropy loss
$$
\min _ {f \in \mathcal {F} _ {\mathbf {L} (\mathcal {X})}} \left\{\widehat {\mathcal {L}} _ {\mathrm {C E}} (f) \triangleq - \sum_ {i = 1} ^ {2} \log \left(\operatorname {s o f t m a x} (f (\mathbf {x} _ {i})) _ {i}\right) \right\}
$$
facilitates satisfaction of Assumption 4.2 with a provably large margin  $\gamma$ . For instance, assuming  $\|f(\mathbf{x})\|_2 \leq \|f(\mathbf{S})\|_2 \leq \beta$  and  $w_{\mathbf{x}} = 1 / |\mathcal{X}|$  for all  $\mathbf{x} \in \mathcal{X}$  (simplified for illustration purposes), for any  $f \in \mathcal{F}_{\mathbf{L}(\mathcal{X})}$  under weak supervision such that
$$
f \in \left\{f \in \mathcal {F} _ {\mathbf {L} (\mathcal {X})} \bigg | \log \left(1 + e ^ {- \sqrt {2} \beta}\right) \leq \widehat {\mathcal {L}} _ {\mathrm {C E}} (f) <   \log \left(1 + e ^ {- \beta}\right) \right\},
$$
Assumption 4.2 is satisfied with
$$
\gamma \geq - \log \left(e ^ {\widehat {\mathcal {L}} _ {\mathrm {C E}} (f)} - 1\right) - \sqrt {2 \beta^ {2} - \log \left(e ^ {\widehat {\mathcal {L}} _ {\mathrm {C E}} (f)} - 1\right) ^ {2}}.
$$
In particular, with the minimum feasible cross-entropy loss  $\widehat{\mathcal{L}}_{\mathrm{CE}}(f) = \log \left(1 + e^{-\sqrt{2}\beta}\right)$ , we have  $\gamma \geq \sqrt{2}\beta$ .
We remark that with DAC regularizations like FixMatch [Sohn et al., 2020] and MixMatch [Berthelot et al., 2019], model predictions are matched against pseudo-labels that are gauged based on data augmentations, usually via the cross-entropy loss. Therefore, DAC has analogous effects on the margins of the learned prediction functions  $f$  as the additional supervision described above.
Moreover, when coupling Equation (1) with the conventional  $KD$  (i.e., feature matching), the learned prediction functions  $f$  inherit the boundedness and margin of the teacher model, which are presumably satisfactory.