SlowGuess/NeurIPS_2023_OCR huggingface dataset:
In the future, a particular theoretical challenge is to propose necessary and sufficient conditions for the BAP and closedness of  $\mathcal{F}_{\mathbf{I}}(\Omega)$ , if possible covering in a single framework both types of domains  $\Omega$  considered here. The fact that the conditions established on these two types of domains are very similar (cf. the similarity between Theorem 3.1 and Theorem 4.1, as well as between Theorem 3.4 and Corollary 4.2) is encouraging. Another interesting algorithmic challenge is to substantially reduce the complexity of the algorithm to decide the closedness of  $\mathcal{L}_{\mathbf{I}}$  in Lemma 3.3, which is currently doubly exponential. It calls for a more efficient algorithm to make this check more practical. Achieving a practically tractable algorithm would for instance allow to check if a support selected e.g. by IMP is pathological or not. This would certainly consolidate the algorithmic robustness and theoretical foundations of pruning techniques to sparsity deep neural networks. From a more theoretical perspective, the existence of an optimum solution in the context of classical linear inverse problems has been widely used to analyze the desirable properties of certain cost functions, e.g.  $\ell^1$  minimization for sparse recovery. Knowing that an optimal solution exists for a given sparse neural network training problem is thus likely to open the door to further fruitful insights.
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This project was supported by the AllegroAssai ANR project ANR-19-CHIA-0009. The authors thank the Blaise Pascal Center (CBP) for the computational means. It uses the SIDUS [27] solution developed by Emmanuel Quemener. Q-T. Le wants to personally thank M-L.Nguyen  $^{3}$  for his enlightenment on the non-equivalence between BAP and closedness in infinite dimension space in Appendix D.
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[5] Tri Dao, Albert Gu, Matthew Eichhorn, Atri Rudra, and Christopher Ré. Learning fast algorithms for linear transforms using butterfly factorizations. In Kamalika Chaudhuri and Ruslan Salakhutdinov, editors, Proceedings of the 36th International Conference on Machine Learning, volume 97 of Proceedings of Machine Learning Research, pages 1517-1527. PMLR, 09-15 Jun 2019.
[6] Tri Dao, Nimit Sharad Sohoni, Albert Gu, Matthew Eichhorn, Amit Blonder, Megan Leszczynski, Atri Rudra, and Christopher Ré. Kaleidoscope: An efficient, learnable representation for all structured linear maps. In 8th International Conference on Learning Representations, ICLR 2020, Addis Ababa, Ethiopia, April 26-30, 2020. OpenReview.net, 2020.
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[8] Steffen Dereich, Arnulf Jentzen, and Sebastian Kassing. On the existence of minimizers in shallow residual relu neural network optimization landscapes, 2023.
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<sup>3</sup>https://sites.google.com/view/mlnguyen/home
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[31] Chaoqi Wang, Guodong Zhang, and Roger Grosse. Picking winning tickets before training by preserving gradient flow. In International Conference on Learning Representations, 2020.
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A Additional notations
In this work, matrices are written in bold uppercase letters. Vectors are written in bold lowercase letters only if they indicate network parameters (such as bias). For a matrix  $\mathbf{A} \in \mathbb{R}^{m \times n}$ , we use  $\mathbf{A}[i,:] \in \mathbb{R}^{1 \times n}$  (resp.  $\mathbf{A}[:,i] \in \mathbb{R}^{m \times 1}$ ) to denote the row (resp. column) vector corresponding the  $i$ th row (resp. column) of  $\mathbf{A}$ . To ease the notation, we write  $\mathbf{A}[i,:]\mathbf{v}$  to denote the scalar product between  $\mathbf{A}[i,:]$  and the vector  $\mathbf{v} \in \mathbb{R}^n$ . This notation will be used regularly when we decompose the functions of one-hidden-neural networks into sum of functions corresponding to hidden neurons.
For a vector  $v \in \mathbb{R}^d$ ,  $v[I] \in \mathbb{R}^{|I|}$  is the vector  $v$  restricted to coefficients in  $I \subseteq [[d]]$ . If  $I = \{i\}$  is a singleton,  $v[i] \in \mathbb{R}$  is the  $i$ th coefficient of  $v$ . We also use  $\mathbf{1}_m$  and  $\mathbf{0}_m$  to denote an all-one (resp. all-zero) vector of size  $m$ .
For a dense (fully connected) feedforward architecture, we denote  $\mathbf{N} = (N_L,\dots ,N_0)$  the dimensions of the input layer  $N_{0} = d$ , hidden layers  $(N_{L - 1},\ldots ,N_{1})$  and output layer  $(N_{L})$ , respectively. The parameters space of the dense architecture  $\mathbf{N}$  is denoted by  $\mathcal{N}_{\mathbf{N}}$ : it is the set of all coefficients of the weight matrices  $\mathbf{W}_i\in \mathbb{R}^{N_i\times N_{i - 1}}$  and bias vectors  $\mathbf{b}_i\in \mathbb{R}^{N_i}$ ,  $i = 1,\dots ,L$ . It is easy to verify that  $\mathcal{N}_{\mathbf{N}}$  is isomorphic to  $\mathbb{R}^N$  where  $N = \sum_{i = 1}^{L}N_{i - 1}N_{i} + \sum_{i = 1}^{L}N_{i}$  is the total number of parameters of the architecture.
Clearly,  $\mathcal{N}_{\mathbf{I}} \subseteq \mathcal{N}_{\mathbf{N}}$  since:
$$
\mathcal {N} _ {\mathbf {I}} := \left\{\theta = \left(\left(\mathbf {W} _ {i}, \mathbf {b} _ {i}\right)\right) _ {i = 1, \dots , L}: \operatorname {s u p p} \left(\mathbf {W} _ {i}\right) \subseteq I _ {i}, \forall i = 1, \dots , L. \right\}. \tag {5}
$$
A special subset of  $\mathcal{N}_{\mathbf{I}}$  is the set of network parameters with zero biases,
$$
\mathcal {N} _ {\mathbf {I}} ^ {0} := \left\{\theta = \left(\left(\mathbf {W} _ {i}, \mathbf {0} _ {N _ {i}}\right)\right) _ {i = 1, \dots , L}: \operatorname {s u p p} \left(\mathbf {W} _ {i}\right) \subseteq I _ {i}, \forall i = 1, \dots , L. \right\}. \tag {6}
$$
Given an activation function  $\nu$ , the realization  $\mathcal{R}_{\theta}^{\nu}$  of a neural network  $\theta \in \mathcal{N}_{\mathbf{N}}$  is the function
$$
\mathcal {R} _ {\theta} ^ {\nu}: x \in \mathbb {R} ^ {N _ {0}} \mapsto \mathcal {R} _ {\theta} ^ {\nu} (x) := \mathbf {W} _ {L} \nu (\dots \nu (\mathbf {W} _ {1} x + \mathbf {b} _ {1}) \dots + \mathbf {b} _ {L - 1}) + \mathbf {b} _ {L} \in \mathbb {R} ^ {N _ {L}} \tag {7}
$$
We denote  $\mathcal{R}^{\nu}:\theta \mapsto \mathcal{R}_{\theta}^{\nu}$  the functional mapping from a set of parameters  $\theta$  to its realization. The function space associated to a sparse architecture  $\mathbf{I}$  and activation function  $\nu$  is the image of  $\mathcal{N}_{\mathbf{I}}$  under  $\mathcal{R}^{\nu}$ :
$$
\mathcal {F} _ {\mathbf {I}} ^ {\nu} := \mathcal {R} ^ {\nu} (\mathcal {N} _ {\mathbf {I}}). \tag {8}
$$
When  $\nu = \sigma$  the ReLU activation function, we recover the definition of realization in Equation (1). We use the shorthands
$$
\mathcal {R} _ {\theta} := \mathcal {R} _ {\theta} ^ {\sigma} \tag {9}
$$
$$
\mathcal {F} _ {\mathbf {I}} := \mathcal {F} _ {\mathbf {I}} ^ {\sigma},
$$
as in the main text. This allows us to define  $\mathcal{L}_{\mathbf{I}}$  (cf. Equation (2)) as  $\mathcal{L}_{\mathbf{I}} \coloneqq \mathcal{R}^{\mathrm{Id}}(\mathcal{N}_{\mathbf{I}}^{0})$  where  $\nu = \mathrm{Id}$  is the identity map, which is a subset of linear maps  $\mathbb{R}^{N_0} \mapsto \mathbb{R}^{N_L}$ .
B Proofs for results in Section 3
Proof. First, we remind the problem of the training of a sparse neural network on a finite data set  $\mathcal{D} = \{(x_i,y_i)\}_{i = 1}^P$ :
$$
\underset {\theta \in \mathcal {N} _ {\mathrm {I}}} {\text {M i n i m i z e}} \quad \mathcal {L} (\theta) := \sum_ {i = 1} ^ {P} \ell \left(\mathcal {R} _ {\theta} \left(x _ {i}\right), y _ {i}\right), \tag {10}
$$
which shares the same optimal value as the following optimization problem:
$$
\underset {\mathbf {D} \in \mathcal {F} _ {\mathbf {I}} (\Omega)} {\text {M i n i m i z e}} \quad \mathcal {L} (\mathbf {D}) := \sum_ {i = 1} ^ {P} \ell \left(\mathbf {D} [:, i ], y _ {i}\right) \tag {11}
$$
where  $\Omega = \{x_{i}\}_{i = 1}^{P}$ . This is simply a change of variables: from  $\mathcal{R}_{\theta}(x_i)$  to the  $i$ th column of  $\mathbf{D} = \mathcal{R}_{\theta}(\Omega)$ . We prove two implications as follows:
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1. Assume the closedness of  $\mathcal{F}_{\mathbf{I}}(\Omega)$  for every finite  $\Omega$ . Then an optimal solution of the optimization problem (10) exists for every finite data set  $\{(x_i, y_i)\}_{i=1}^P$ . Consider a training set  $\{(x_i, y_i)\}_{i=1}^P$  and  $\Omega := \{x_i\}_{i=1}^P$ . Since  $\mathbf{D} := \mathbf{0}_{P \times N_L} \in \mathcal{F}_{\mathbf{I}}(\Omega)$  (by setting all parameters in  $\theta$  equal to zero), the set  $\mathcal{F}_{\mathbf{I}}(\Omega)$  is non-empty. The optimal value of (11) is thus upper bounded by  $\mathcal{L}(\mathbf{0})$ . Since the function  $\ell(\cdot, y_i)$  is coercive for every  $y_i$  in the training set, there exists a constant  $C$  (dependent on the training set and the loss) such that minimizing (11) on  $\mathcal{F}_{\mathbf{I}}(\Omega)$  or on  $\mathcal{F}_{\mathbf{I}}(\Omega) \cap \mathcal{B}(0, C)$  (with  $\mathcal{B}(\mathbf{0}, C)$  the  $L^2$  ball of radius  $C$  centered at zero) yields the same infimum. The function  $\mathcal{L}$  is continuous, since each  $\ell(\cdot, y_i)$  is continuous by assumption, and the set  $\mathcal{F}_{\mathbf{I}}(\Omega) \cap \mathcal{B}(0, C)$  is compact, since it is closed (as an intersection of two closed sets) and bounded (since  $\mathcal{B}(0, C)$  is bounded). As a result there exists a matrix  $\mathbf{D} \in \mathcal{F}_{\mathbf{I}}(\Omega) \cap \mathcal{B}(0, C)$  yielding the optimal value for (11). Thus, the parameters  $\theta$  such that  $\mathcal{R}_{\theta}(\Omega) = \mathbf{D}$  is an optimal solution of (10).
2. Assume that an optimal solution of problem 10 exists for every finite data set  $\{(x_i, y_i)\}_{i=1}^P$ . Then  $\mathcal{F}_{\mathbf{I}}(\Omega)$  is closed for every  $\Omega$  finite. We prove the contraposition of this claim. Assume there exists a finite set  $\Omega = \{x_i\}_{i=1}^P$  such that  $\mathcal{F}_{\mathbf{I}}(\Omega)$  is not closed. Then, there exists a matrix  $\mathbf{D} \in \mathbb{R}^{N_L \times P}$  such that  $\mathbf{D} \in \overline{\mathcal{F}_{\mathbf{I}}(\Omega)} \setminus \mathcal{F}_{\mathbf{I}}(\Omega)$ . Consider the dataset  $\{(x_i, y_i)\}_{i=1}^P$  where  $y_i \in \mathbb{R}^{N_L}$  is the  $i$ th column of  $\mathbf{D}$ . We prove that the infimum value of (10) is  $V := \sum_{i=1}^P \ell(y_i, y_i)$ . Indeed, since  $\mathbf{D} \in \overline{\mathcal{F}_{\mathbf{I}}(\Omega)}$ , there exists a sequence  $\{\theta_k\}_{k \in \mathbb{N}}$  such that  $\lim_{k \to \infty} \mathcal{R}_{\theta_k}(\Omega) = \mathbf{D}$ . Therefore, by continuity of  $\ell(\cdot, y_i)$ , we have:
$$
\lim  _ {k \rightarrow \infty} \mathcal {L} \left(\theta_ {k}\right) = \sum_ {i = 1} ^ {P} \lim  _ {k \rightarrow \infty} \ell \left(\mathcal {R} _ {\theta_ {k}} \left(x _ {i}\right), y _ {i}\right) = \sum_ {i = 1} ^ {P} \ell \left(y _ {i}, y _ {i}\right) = V.
$$
Moreover, the infimum cannot be smaller than  $V$  because the  $i$ th summand is at least  $\ell(y_i, y_i)$  (due to the assumption on  $\ell$  in Proposition 3.1). Therefore, the infimum value is indeed  $V$ . Since we assume that  $y$  is the only minimizer of  $y' \mapsto \ell(y', y)$ , this value can be achieved only if there exists a parameter  $\theta \in \mathbf{I}$  such that  $\mathcal{R}_{\theta}(\Omega) = \mathbf{D}$ . This is impossible due to our choice of  $\mathbf{D}$  which does not belong to  $\mathcal{F}_{\mathbf{I}}(\Omega)$ . We conclude that with our constructed data set  $\mathcal{D}$ , an optimal solution does not exist for (10).
B.2 Proof of Lemma 3.2
The proof of Lemma 3.2 (and thus, as discussed in the main text, of Theorem 3.1) use four technical lemmas. Lemma B.1 is proved in Appendix C.1 since it involves Theorem 4.1. The other lemmas are proved right after the proof of Lemma 3.2.
Lemma B.1. If  $\mathbf{A} \in \overline{\mathcal{L}_{\mathbf{I}}} \backslash \mathcal{L}_{\mathbf{I}} \subseteq \mathbb{R}^{N_L \times N_0}$  then the function  $f: x \mapsto f(x) \coloneqq \mathbf{Ax}$  satisfies  $f \in \overline{\mathcal{F}_{\mathbf{I}}(\Omega)} \setminus \mathcal{F}_{\mathbf{I}}(\Omega)$  for every subset  $\Omega$  of  $\mathbb{R}^{N_0}$  that is bounded with non-empty interior.
Lemma B.2. Consider  $\Omega = \{x_i\}_{i=1}^P$  a finite subset of  $\mathbb{R}^d$  and  $\Omega' = [-B, B]^d$  such that  $\Omega \subseteq \Omega'$ . If  $f \in \overline{\mathcal{F}_{\mathbf{I}}(\Omega')} \text{ (under the topology induced by } \| \cdot \|_{\infty})$ , then  $\mathbf{D} := [f(x_1) \ldots f(x_P)] \in \overline{\mathcal{F}_{\mathbf{I}}(\Omega)}$ .
Lemma B.3. Consider  $\mathcal{R}_{\theta}$ , the realization of a ReLU neural network with parameter  $\theta \in \mathbf{I}$ . This function is continuous and piecewise linear. On the interior of each piece, its Jacobian matrix is constant and satisfies  $\mathbf{J} \in \mathcal{L}_{\mathbf{I}}$ .
Lemma B.4. For  $p, N \in \mathbb{N}$ , consider the following set of points (a discretized grid for  $[0,1]^N$ ):
$$
\Omega = \Omega_ {p} ^ {N} = \left\{\left(\frac {i _ {1}}{p}, \dots , \frac {i _ {N}}{p}\right) \mid 0 \leq i _ {j} \leq p, i _ {j} \in \mathbb {N}, \forall 1 \leq j \leq N \right\}.
$$
If  $H \in \mathbb{N}$  satisfies  $p \geq 3NH$ , then for any collection of  $H$  hyperplanes, there exists  $x \in \Omega_p^N$  such that the elementary hypercube whose vertices are of the form
$$
\left\{x + \left(\frac {i _ {1}}{p}, \dots , \frac {i _ {N}}{p}\right) \mid i _ {j} \in \{0, 1 \} \forall 1 \leq j \leq N \right\} \subseteq \Omega_ {p} ^ {N}
$$
lies entirely inside a polytope delimited by these hyperplanes.
We are now ready to prove Lemma 3.2.
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Proof of Lemma 3.2. Since  $\mathcal{L}_{\mathbf{I}}$  is not closed, there exists a matrix  $\mathbf{A} \in \overline{\mathcal{L}_{\mathbf{I}}} \setminus \mathcal{L}_{\mathbf{I}}$ , and we consider  $f(x) \coloneqq \mathbf{A}x$ . Setting  $p \coloneqq 3N_04^{\sum_{i=1}^{L-1} N_i}$  we construct  $\Omega$  as the grid:
$$
\Omega = \left\{\left(\frac {i _ {1}}{p}, \dots , \frac {i _ {N _ {0}}}{p}\right) \mid 0 \leq i _ {j} \leq p, i _ {j} \in \mathbb {N}, \forall 1 \leq j \leq N _ {0} \right\},
$$
so that the cardinality of  $\Omega = \{x_{i}\}_{i = 1}^{P}$  is  $P\coloneqq (p + 1)^{N_0}$ . Similar to the sketch proof, consider  $\mathbf{D}\coloneqq \left[f(x_1),f(x_2),\ldots ,f(x_P)\right]$ . Our goal is to prove that  $\mathbf{D}\in \overline{\mathcal{F}_{\mathbf{I}}(\Omega)}\setminus \mathcal{F}_{\mathbf{I}}(\Omega)$ .
First, notice that  $\mathbf{D} \in \overline{\mathcal{F}_{\mathbf{I}}(\Omega)}$  as an immediate consequence of Lemma B.2 and Lemma B.1.
It remains to show that  $\mathbf{D} \notin \mathcal{F}_{\mathbf{I}}(\Omega)$ . We proceed by contradiction, assuming that there exists  $\theta \in \mathcal{N}_{\mathbf{I}}$  such that  $\mathcal{R}_{\theta}(\Omega) = \mathbf{D}$ .
To show the contradiction, we start by showing that, as a consequence of Lemma B.4 there exists  $x \in \Omega$  such that the hypercube whose vertices are the  $2^{N_0}$  points
$$
\left\{x + \left(\frac {i _ {1}}{p}, \dots , \frac {i _ {N _ {0}}}{p}\right) \mid i _ {j} \in \{0, 1 \}, \forall 1 \leq j \leq N _ {0} \right\} \subseteq \Omega , \tag {12}
$$
lies entirely inside a linear region  $\mathcal{P}$  of the continuous piecewise linear function  $\mathcal{R}_{\theta}$  [1]. Denote  $K = 2^{\sum_{i=1}^{L} N_i}$  a bound on the number of such linear regions, see e.g. [24]. Each frontier between a pair of linear regions can be completed into a hyperplane, leading to at most  $H = K^2$  hyperplanes. Since  $p = 3N_0K^2 \geq 3N_0H$ , by Lemma B.4 there exists  $x \in \Omega$  such that the claimed hypercube lies entirely inside a polytope delimited by these hyperplanes. As this polytope is itself included in some linear region  $\mathcal{P}$  of  $\mathcal{R}_{\theta}$ , this establishes our intermediate claim.
Now, define  $v_{0} \coloneqq x$  and  $v_{i} \coloneqq x + (1 / p)\mathbf{e}_{i}, i \in [[N_{0}]]$  where  $\mathbf{e}_i$  is the  $i$ th canonical vector. Denote  $\mathbf{P} \in \mathbb{R}^{N_L \times N_0}$  the matrix such that the restriction of  $\mathcal{R}_{\theta}$  to the piece  $\mathcal{P}$  is  $f_{\mathcal{P}}(x) = \mathbf{P}x + \mathbf{b}$ . Since  $\mathbf{P}$  is the Jacobian matrix of  $\mathcal{R}_{\theta}$  in the linear region  $\mathcal{P}$ , we deduce from Lemma B.3 that  $\mathbf{P} \in \mathcal{L}_{\mathbf{I}}$ . Since the points  $v_{i}$  belong to the hypercube which is both included in  $\mathcal{P}$  and in  $\Omega$  we have for each  $i$ :
$$
\begin{array}{l} \mathbf {P} \left(v _ {0} - v _ {i}\right) = f _ {\mathcal {P}} \left(v _ {0}\right) - f _ {\mathcal {P}} \left(v _ {i}\right) \\ = \mathcal {R} _ {\theta} (v _ {0}) - \mathcal {R} _ {\theta} (v _ {i}) \\ = f \left(v _ {0}\right) - f \left(v _ {i}\right) \\ = \mathbf {A} (v _ {0} - v _ {i}). \\ \end{array}
$$
where the third equality follows from the definition of  $\mathbf{D}$  and the fact that we assume  $\mathcal{R}_{\theta}(\Omega) = \mathbf{D}$ . Since  $v_{0} - v_{i} = e_{i} / p, i = 1,\dots ,n$  are linearly independent, we conclude that  $\mathbf{P} = \mathbf{A}$ . This implies  $\mathbf{A}\in \mathcal{L}_{\mathbf{I}}$ , hence the contradiction. This concludes the proof.
We now prove the intermediate technical lemmas.
Proof of Lemma B.2. Since  $f \in \overline{\mathcal{F}_{\mathbf{I}}(\Omega')}$ , there exists a sequence  $\{\theta_k\}_{k \in \mathbb{N}}$  such that:
$$
\lim  _ {k \rightarrow \infty} \sup  _ {x \in \Omega^ {\prime}} \| f (x) - \mathcal {R} _ {\theta_ {k}} (x) \| = 0
$$
Denoting  $\mathbf{D}_k \coloneqq \left[\mathcal{R}_{\theta_k}(x_1) \dots \mathcal{R}_{\theta_k}(x_r)\right]$ , since  $x_i \in \Omega \subseteq \Omega'$ ,  $i = 1, \ldots, P$ , it follows that  $\mathbf{D}_k$  converges to  $\mathbf{D}$ . Since  $\mathbf{D}_k \in \mathcal{F}_{\mathbf{I}}(\Omega)$  by construction, we get that  $\mathbf{D} \in \overline{\mathcal{F}_{\mathbf{I}}(\Omega)}$ .
Proof of Lemma B.3. For any  $\theta \in \mathbf{I}$ ,  $\mathcal{R}_{\theta}$  is a continuous piecewise linear function since it is the realization of a ReLU neural network [1]. Consider  $\mathcal{P}$  a linear region of  $\mathcal{R}_{\theta}$  with non-empty interior. The Jacobian matrix of  $\mathcal{P}$  has the following form [28, Lemma 9]:
$$
\mathbf {J} = \mathbf {W} _ {L} \mathbf {D} _ {L - 1} \mathbf {W} _ {L - 1} \mathbf {D} _ {L - 2} \dots \mathbf {D} _ {1} \mathbf {W} _ {1}
$$
where  $\mathbf{D}_i$  is a binary diagonal matrix (diagonal matrix whose coefficients are either one or zero).
Since  $\operatorname{supp}(\mathbf{D}_i\mathbf{W}_i) \subseteq \operatorname{supp}(\mathbf{W}_i) \subseteq I_i$ , we have:  $\mathbf{J} = \mathbf{W}_L \prod_{i=1}^{L-1} (\mathbf{D}_i\mathbf{W}_i) \in \mathcal{L}_I$ .
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Proof of Lemma B.4. Every edge of an elementary hypercube can be written as:
$$
\left(x, x + \frac {1}{p} \mathbf {e} _ {i}\right), x \in \Omega_ {p} ^ {N}
$$
where  $\mathbf{e}_i$  is the  $i$ th canonical vector,  $1 \leq i \leq N$ . The points  $x$  and  $x + (1/p)\mathbf{e}_i$  are two endpoints. Note that in this proof we use the notation  $(a,b)$  to denote the line segment whose endpoints are  $a$  and  $b$ . By construction,  $\Omega_p^N$  contains  $p^N$  such elementary hypercubes. Given a collection of  $H$  hyperplanes, we say that an elementary hypercube is an intersecting hypercube if it does not lie entirely inside a polytope generated by the hyperplanes, meaning that there exists a hyperplane that intersects at least one of its edges. More specifically, an edge and a hyperplane intersect if they have exactly one common point. We exclude the case where there are more than two common points since that implies that the edge lies completely in the hyperplane. The edges that are intersected by at least one hyperplane are called intersecting edges. Note that a hypercube can have intersecting edges, but it may not be an intersecting one. A visual illustration of this idea is presented in Figure 3.
b)
Figure 3: Illustration of definitions in  $\mathbb{R}^2$ : a) an intersecting hypercube with two intersecting edges; b) not an intersecting hypercube, but it has two intersecting edges; c) not an intersecting hypercube and it only has two intersecting edges (not three according to our definitions: the bottom edge is not intersecting).
Formally, a hyperplane  $\{w^\top x + b = 0\}$  for  $w \in \mathbb{R}^N$  and  $b \in \mathbb{R}$  intersects an edge  $(x, x + \frac{1}{p}\mathbf{e}_i)$  if:
$$
\left\{ \begin{array}{l} \left(w ^ {\top} x + b\right) \left[ w ^ {\top} \left(x + \frac {1}{p} \mathbf {e} _ {i}\right) + b \right] \leq 0 \\ \text {a n d} \\ w ^ {\top} x + b \neq 0 \text {o r} w ^ {\top} \left(x + \frac {1}{p} \mathbf {e} _ {i}\right) + b \neq 0 \end{array} \right. \tag {13}
$$
We further illustrate these notions in Figure 4. We emphasize that according to Equation (13),  $\ell_3$  in Figure 4 does not intersect any edge along its direction.