SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Definition A.3. Given a dataset  $X = \{x_{1},\ldots ,x_{n}\} \subset \mathbb{R}^{d}$ , as well as parameters  $k\in \mathbb{N}$ ,  $z\geq 1$  and  $\varepsilon >0$ , a (strong)  $\varepsilon$ -coreset for  $(k,z)$ -clustering is specified by a set of points  $Y\subset \mathbb{R}^d$  and a weight function  $w\colon Y\to \mathbb{R}_{\geq 0}$ , such that, for every set  $C = \{c_1,\dots ,c_k\} \subset \mathbb{R}^d$ ,
$$
(1 - \varepsilon) \cdot \operatorname {c o s t} _ {z} (X, C) \leq \sum_ {y \in Y} w (y) \cdot \min  _ {j \in [ k ]} \| y - c _ {j} \| _ {2} ^ {z} \leq (1 + \varepsilon) \cdot \operatorname {c o s t} _ {z} (X, C).
$$
Coresets are constructed via "sensitivity sampling," a technique that, given an approximate clustering of a dataset  $X$ , produces a probability distribution such that sampling enough points from this distribution results in a coreset.
Definition A.4 (Sensitivity Sampling). Consider a dataset  $X = \{x_{1},\ldots ,x_{n}\} \subset \mathbb{R}^{d}$ , as well as parameters  $k\in \mathbb{N}$ $z\geq 1$ . For a centet set  $C = \{c_1,\dots ,c_k\} \subset \mathbb{R}^d$  and assignment  $\sigma \colon [n]\to [k]$ , let  $X_{j} = \{x_{i}:\sigma (i) = j\}$ . We let  $\mathcal{D}$  be a distribution supported on  $X$  where
$$
\operatorname * {P r} _ {\boldsymbol {x} \sim \mathcal {D}} \left[ \boldsymbol {x} = x _ {i} \right] \propto \frac {\| x _ {i} - c _ {\sigma (i)} \| _ {2} ^ {2}}{\sum_ {j = 1} ^ {n} \| x _ {j} - c _ {\sigma (j)} \| _ {2} ^ {2}} + \frac {1}{| X _ {\sigma (i)} |}.
$$
The main theorem that we will use is given below, which shows that given a center set and an assignment that gives an  $\alpha$ -approximation to  $(k,z)$ -clustering, one may sample from the distribution  $\mathcal{D}$  defined about in order to generate a coreset with high probability.
Theorem A.2 ([15]). For any dataset  $X = \{x_{1},\ldots ,x_{n}\} \subset \mathbb{R}^{d}$  and any parameters  $k\in \mathbb{N}$  and  $z\geq 1$ , suppose that  $C = \{c_1,\dots ,c_k\} \subset \mathbb{R}^d$  and  $\sigma \colon [n]\to [k]$  is a  $\alpha$ -approximation to  $(k,z)$ -clustering, i.e.,
$$
\sum_ {i = 1} ^ {n} \left\| x _ {i} - c _ {\sigma (i)} \right\| _ {2} ^ {z} \leq \alpha \operatorname {o p t} _ {k, z} (X).
$$
Letting  $\mathcal{D}$  denote the distribution specified in Definition A.4, the following occurs with high probability.
- We let  $\mathbf{y}_1, \ldots, \mathbf{y}_s$  denote independent samples from  $\mathcal{D}$ , and  $w(\mathbf{y}_i)$  be the inverse of the probability that  $\mathbf{y}_i$  is sampled according to  $\mathcal{D}$ . We set  $s \geq \mathrm{poly}(kd \cdot \alpha \cdot 2^z / \varepsilon)$ .
- The set  $Y = \{y_{1},\ldots ,y_{s}\}$  with weights  $w$  is an  $\varepsilon$ -coreset for  $(k,z)$ -clustering.
A.3 A Simple Lemma
We will repeatedly use the following simple lemma.
Lemma A.3. Let  $a, b \in \mathbb{R}_{\geq 0}$  be any two numbers and  $z \geq 1$ . Then,  $(a + b)^z \leq 2^{z - 1}a^z + 2^{z - 1}b^z$ .
Proof. The function  $\phi(t) = t^z$  is convex for  $z \geq 1$ , so Jensen's inequality implies  $\phi((a + b)/2) \leq (1/2)\phi(a) + (1/2)\phi(b)$ .
B Approximation Guarantees from One-Dimensional Projections
In this section, we prove Theorem 2.2 (rather, the generalization of Theorem 2.2 to any  $z \geq 1$ ) by analyzing the random one-dimensional projection step in our algorithm. In order to introduce some notation, let  $X = \{x_{1}, \ldots, x_{n}\} \subset \mathbb{R}^{d}$  be a set of points, and for a partition of  $X$  into  $k$  sets,  $(Y_{1}, \ldots, Y_{k})$ , we let the  $(k, z)$ -clustering cost of  $X$  with the partition  $(Y_{1}, \ldots, Y_{k})$  be
$$
\operatorname {c o s t} _ {z} \left(Y _ {1}, \dots , Y _ {k}\right) = \sum_ {i = 1} ^ {k} \min  _ {c \in \mathbb {R} ^ {d}} \sum_ {x \in Y _ {i}} \| x - c \| _ {2} ^ {z} \tag {2}
$$
and call the  $k$  points  $c$  selected as minima a set of centers realizing the  $(k,z)$ -clustering cost of  $(Y_{1},\ldots ,Y_{k})$ . We note that (2) is a cost function for  $(k,z)$ -clustering, but it is different from
Definition A.1. In Definition A.1, the emphasis is on the set of  $k$  centers  $C = \{c_1,\dots ,c_k\}$ , and the induced set of clustering of  $X$ , i.e., the partition  $(Y_{1},\ldots ,Y_{k})$  given by assigning points to the closest center, is only implicitly specified by the set of centers. On the other hand, (2) emphasizes the clustering  $(Y_{1},\ldots ,Y_{k})$ , and the set of  $k$  centers implicitly specified by  $(Y_{1},\ldots ,Y_{k})$ . The optimal set of centers and the optimal clustering will achieve the same cost; however, our proof will mostly consider the clustering  $(Y_{1},\ldots ,Y_{k})$  as the object to optimize. Shortly, we will sample a (random) dimensionality reduction map  $\Pi \colon \mathbb{R}^d\to \mathbb{R}^t$  and seek bounds for  $t = 1$ . We will write  $\mathrm{cost}_z(\Pi (Y_1),\dots,\Pi (Y_k))$  for the cost of clustering the points after applying the dimensionality reduction map  $\Pi$  to the partition  $Y_{1},\ldots ,Y_{k}$ . Namely, we write
$$
\operatorname {c o s t} _ {z} \left(\Pi \left(Y _ {1}\right), \dots , \Pi \left(Y _ {k}\right)\right) = \sum_ {i = 1} ^ {k} \min  _ {c \in \mathbb {R} ^ {t}} \sum_ {x \in Y _ {i}} \| \Pi (x) - c \| _ {2} ^ {z}.
$$
Definition B.1. For a set of points  $X = \{x_{1},\ldots ,x_{n}\} \subset \mathbb{R}^{d}$ , we use  $X_{1}^{*},\ldots ,X_{k}^{*}$  of  $X$  to denote the partition of  $X$  with minimum  $(k,z)$ -clustering cost and we use  $C^* = \{c_1^*,\dots ,c_k^*\} \subset \mathbb{R}^d$  to denote a set of  $k$  centers which realizes the  $(k,z)$ -clustering cost of  $X_{1}^{*},\ldots ,X_{k}^{*}$ , i.e., the set of centers which satisfies
$$
\operatorname {c o s t} _ {z} \left(X _ {1} ^ {*}, \dots , X _ {k} ^ {*}\right) = \sum_ {i = 1} ^ {k} \sum_ {x \in X _ {i} ^ {*}} \| x - c _ {i} ^ {*} \| _ {2} ^ {z}.
$$
By slight abuse of notation, we also let  $c^* \colon X \to C^*$  be the map which sends every point of  $X$  to its corresponding center (i.e., if  $x \in X_i^*$ , then  $c^*(x)$  is the point  $c_i^*$ ).
We prove the following lemma, which generalizes Lemma 2.5 from  $k$ -means to  $(k,z)$ -clustering (recall that  $k$ -means corresponds to the case of  $z = 2$ ).
Lemma B.1 (Effect of One-Dimensional Projection on  $(k,z)$ -Clustering). For  $n,d,k\in \mathbb{N}$  and  $z\geq 1$ , let  $X = \{x_{1},\ldots ,x_{n}\} \subset \mathbb{R}^{d}$  be an arbitrary dataset. We consider the (random) linear map  $\Pi \colon \mathbb{R}^d\to \mathbb{R}$  given by sampling  $\mathbf{g}\sim \mathcal{N}(0,I_d)$  and setting
$$
\boldsymbol {\Pi} (\boldsymbol {x}) = \langle \boldsymbol {x}, \mathbf {g} \rangle .
$$
With probability at least 0.9 over  $\mathbf{g}$ , the following occurs:
- We consider the projected dataset  $\mathbf{X}' = \{\pmb{x}_1',\dots,\pmb{x}_n'\} \subset \mathbb{R}$  be given by  $\pmb{x}_i' = \pmb{\Pi}(x_i)$ , and
- For any  $r \geq 1$ , we let  $(Y_1, \ldots, Y_k)$  denote any partition of  $X$  satisfying
$$
\operatorname {c o s t} _ {z} \left(\Pi \left(Y _ {1}\right), \dots , \Pi \left(Y _ {k}\right)\right) \leq r \cdot \min  _ {c _ {1}, \dots , c _ {k} \in \mathbb {R}} \sum_ {i = 1} ^ {n} \min  _ {j \in [ k ]} \left| \boldsymbol {x} _ {i} ^ {\prime} - c _ {j} \right| ^ {z}.
$$
$$
\operatorname {c o s t} _ {z} \left(Y _ {1}, \dots , Y _ {k}\right) \leq 2 ^ {O (z)} \cdot k ^ {2 z} \cdot r \cdot \operatorname {o p t} _ {k, z} (X).
$$
By setting  $z = 2$ , we obtain the desired bound from Lemma 2.5. We can immediately see that, from Lemma B.1, and the approximation guarantees of  $k$ -means++ (or rather, its generalization to  $z \geq 1$ ) in Theorem A.1, we obtain our desired approximation guarantees. Below, we state the generalization of Theorem 2.2 to all  $z \geq 1$  and, assuming Lemma B.1, its proof.
Theorem B.2 (Generalization of Theorem 2.2 to  $z \geq 1$ ). For  $n, d, k \in \mathbb{N}$  and  $z \geq 1$ , let  $X = \{x_1, \ldots, x_n\} \subset \mathbb{R}^d$  be an arbitrary dataset. We consider the following generalization of our algorithm PRONE:
1. Sample a random Gaussian vector  $\mathbf{g} \sim \mathcal{N}(0, I_d)$  and consider the projection  $\mathbf{X}' = \{\mathbf{x}_1', \ldots, \mathbf{x}_n'\}$  given by  $\mathbf{x}_i' = \boldsymbol{\Pi}(x_i)$ , for  $\boldsymbol{\Pi}(x) = \langle x, \mathbf{g} \rangle \in \mathbb{R}$ .
2. Execute the (generalization of the)  $k$ -means++ seeding strategy for  $z \geq 1$  of Definition A.2 with the dataset  $\mathbf{X}' \subset \mathbb{R}$ , and let  $\mathbf{x}_{j_1}', \ldots, \mathbf{x}_{j_k}' \in \mathbb{R}$  denote the centers and  $(Y_1, \ldots, Y_k)$  denote the partition of  $X$  specifying the  $k$  clusters found.
3. Output the clustering  $(\mathbf{Y}_1, \ldots, \mathbf{Y}_k)$ , and the set of centers  $\mathbf{c}_1, \ldots, \mathbf{c}_k \in \mathbb{R}^d$  where
$$
\pmb {c} _ {\ell} = \underset {\pmb {x} \sim \pmb {Y} _ {\ell}} {\mathbb {E}} \left[ \pmb {x} \right] \in \mathbb {R} ^ {d}.
$$
Then, with probability at least 0.8 over the execution of the algorithm,
$$
\operatorname {c o s t} _ {z} \left(\boldsymbol {Y} _ {1}, \dots , \boldsymbol {Y} _ {k}\right) \leq \sum_ {\ell = 1} ^ {k} \sum_ {x \in Y _ {\ell}} \| x - \boldsymbol {c} _ {\ell} \| _ {2} ^ {z} \leq 2 ^ {O (z)} \cdot k ^ {2 z} \cdot \log k \cdot \operatorname {o p t} _ {k, z} (X).
$$
Proof of Theorem B.2 assuming Lemma B.1. We consider the case (over the randomness in the execution of the algorithm) that:
1. The conclusions of Lemma B.1 hold for the projected dataset  $\mathbf{X}'$  (which happens with probability at least 0.9) by Lemma B.1.
2. The execution of the generalization  $k$ -means++ seeding strategy on  $\mathbf{X}'$  (from Definition A.2) produces a set of centers  $\{\pmb{x}_{j_1}',\dots,\pmb{x}_{j_k}'\} \subset \mathbb{R}$  which cluster  $\mathbf{X}'$  with cost at most  $O(2^{2z}\log k)\cdot \mathrm{opt}_{k,z}(\mathbf{X}')$  (which also happens with probability 0.9 by Markov's inequality).
By a union bound, both hold with probability at least 0.8. We now use Lemma B.1 to upper bound the cost of the clustering  $(\mathbf{Y}_1,\dots ,\mathbf{Y}_k)$ . The first inequality is trivial; suppose we let  $\hat{c}_1^*,\ldots ,\hat{c}_\ell^*\in \mathbb{R}^d$  be the centers which minimize for each  $\ell \in [k]$
$$
\min  _ {\hat {c} _ {\ell} \in \mathbb {R} ^ {d}} \sum_ {x \in \boldsymbol {Y} _ {\ell}} \| x - \hat {c} _ {\ell} \| _ {2} ^ {z} = \sum_ {x \in \boldsymbol {Y} _ {\ell}} \| x - \hat {c} _ {\ell} ^ {*} \| _ {2} ^ {z}.
$$
Then, we trivially have
$$
\operatorname {c o s t} _ {z} \left(\boldsymbol {Y} _ {1}, \dots , \boldsymbol {Y} _ {k}\right) = \sum_ {\ell = 1} ^ {k} \sum_ {x \in \boldsymbol {Y} _ {\ell}} \| x - \hat {c} _ {\ell} ^ {*} \| _ {2} ^ {z} \leq \sum_ {\ell = 1} ^ {k} \sum_ {x \in \boldsymbol {Y} _ {\ell}} \| x - \boldsymbol {c} _ {\ell} \| _ {2} ^ {z}.
$$
Furthermore, we can also show a corresponding upper bound. For each  $\ell \in [k]$ , recall that  $c_{\ell} \in \mathbb{R}^{d}$  is the center of mass of  $\pmb{Y}_{k}$ , so we can apply the triangle inequality and Lemma A.3
$$
\begin{array}{l} \sum_ {x \in \mathbf {Y} _ {\ell}} \| x - \boldsymbol {c} _ {\ell} \| _ {2} ^ {z} \leq 2 ^ {z - 1} \sum_ {x \in \mathbf {Y} _ {\ell}} \| x - \hat {c} _ {\ell} ^ {*} \| _ {2} ^ {z} + 2 ^ {z - 1} | \mathbf {Y} _ {\ell} | \cdot \| \hat {c} _ {\ell} ^ {*} - \underset {\boldsymbol {x} \sim \boldsymbol {Y} _ {\ell}} {\mathbb {E}} [ \boldsymbol {x} ] \| _ {2} ^ {z} \\ \leq 2 ^ {z - 1} \sum_ {x \in \boldsymbol {Y} _ {\ell}} \| x - \hat {c} _ {\ell} ^ {*} \| _ {2} ^ {z} + 2 ^ {z - 1} | \boldsymbol {Y} _ {\ell} | \cdot \underset {\boldsymbol {x} \sim \boldsymbol {Y} _ {\ell}} {\mathbb {E}} \left[ \| x - \hat {c} _ {\ell} ^ {*} \| _ {2} ^ {z} \right], \\ \end{array}
$$
where the second inequality is Jensen's inequality, since  $\phi (x) = \| \hat{c}_\ell^* -x\| _2^z$  is convex for  $z\geq 1$ . Thus, we have upper-bounded
$$
\sum_ {x \in \boldsymbol {Y} _ {\ell}} \| x - \boldsymbol {c} _ {\ell} \| _ {2} ^ {z} \leq 2 ^ {z} \sum_ {x \in \boldsymbol {Y} _ {\ell}} \| x - \hat {c} _ {\ell} ^ {*} \| _ {2} ^ {z},
$$
$$
\sum_ {\ell = 1} ^ {k} \sum_ {x \in \mathbf {Y} _ {\ell}} \| x - \mathbf {c} _ {\ell} \| _ {2} ^ {z} \leq 2 ^ {z} \cdot \operatorname {c o s t} _ {z} \left(\mathbf {Y} _ {1}, \dots , \mathbf {Y} _ {k}\right). \tag {3}
$$
The final step involves relating  $\mathrm{cost}_z(\mathbf{Y}_1,\ldots ,\mathbf{Y}_k)$  using the conclusions of Lemma B.1. Notice that our algorithm produces the clustering  $(\mathbf{Y}_1,\dots ,\mathbf{Y}_k)$  of  $\pmb{X}^{\prime}$  which is specified by letting
$$
\boldsymbol {Y} _ {\ell} = \left\{x _ {i} \in X: \forall j ^ {\prime} \in [ k ], | \boldsymbol {x} _ {i} ^ {\prime} - \boldsymbol {x} _ {j _ {\ell}} ^ {\prime} | \leq | \boldsymbol {x} _ {i} ^ {\prime} - \boldsymbol {x} _ {j ^ {\prime}} | ^ {z} \right\},
$$
and by the event (2), we have  $\mathrm{cost}_z(\Pi(\mathbf{Y}_1), \ldots, \Pi(\mathbf{Y}_k)) \leq O(2^{2z} \log k) \cdot \mathrm{opt}_{k,z}(\mathbf{X}')$ . By event (1), Lemma B.1 implies that  $\mathrm{cost}_z(\mathbf{Y}_1, \ldots, \mathbf{Y}_k) \leq 2^{O(z)} \cdot k^{2z} \cdot O(2^{2z} \log k) \cdot \mathrm{opt}_{k,z}(X)$ . Combined with (3), we obtain our desired bound.
B.1 Proof of Lemma B.1
We now turn to the proof of Lemma B.1, where our analysis will proceed in two steps. First, we assume a fixed dimensionality reduction map  $\Pi \colon \mathbb{R}^d \to \mathbb{R}^t$ , which satisfies two geometrical conditions on  $\Pi$ . Under these conditions, we show how to "lift" an approximate clustering of the mapped points in  $\mathbb{R}^t$  to an approximate clustering of the original dataset in  $\mathbb{R}^d$  at the cost of weakening the approximation ratio. Then, we show that a simple one-dimensional projection  $\Pi \colon \mathbb{R}^d \to \mathbb{R}$  given by  $\Pi(x) = \langle x, \mathbf{g} \rangle$ , for  $\mathbf{g}$  being sampled from a  $d$ -dimensional standard Gaussian, satisfies the geometrical conditions of our lemma.
Lemma B.3. Let  $X = \{x_{1},\ldots ,x_{n}\} \subset \mathbb{R}^{d}$  and  $\Pi \colon \mathbb{R}^d\to \mathbb{R}^t$  be a linear map. Let  $C = \{c_1^*,\dots ,c_k^*\} \subset \mathbb{R}^d$  denote the set of centers minimizing  $\mathrm{cost}_z(X,C)$ , and  $(X_1^*,\dots ,X_k^*)$  denote the optimal  $(k,z)$ -clustering, and suppose that for the parameters  $\mathsf{D}_1,\mathsf{D}_2,\mathsf{D}_3\geq 1$ , the following conditions hold:
- Centers Don't Contract: Every  $i, j \in [k]$  satisfies
$$
\left\| c _ {i} ^ {*} - c _ {j} ^ {*} \right\| _ {2} \leq \mathrm {D} _ {1} \cdot \left\| \Pi \left(c _ {i} ^ {*}\right) - \Pi \left(c _ {j} ^ {*}\right) \right\| _ {2}.
$$
- Cost of  $(X_1^*,\ldots ,X_k^*)$  does not Increase: We have that
$$
\sum_ {i = 1} ^ {k} \sum_ {x \in X _ {i} ^ {*}} \| \Pi (x) - \Pi \left(c _ {i} ^ {*}\right) \| _ {2} ^ {z} \leq \mathrm {D} _ {2} \cdot \operatorname {c o s t} _ {z} \left(X _ {1} ^ {*}, \dots , X _ {k} ^ {*}\right).
$$
- Approximately Optimal  $(\Pi(Y_1), \ldots, \Pi(Y_k))$ : The partition  $(Y_1, \ldots, Y_k)$  of  $X$  is  $\mathsf{D}_3$ -approximately optimal for  $\Pi(X)$ , i.e.,
$$
\operatorname {c o s t} _ {z} \left(\Pi \left(Y _ {1}\right), \dots , \Pi \left(Y _ {k}\right)\right) \leq \mathrm {D} _ {3} \cdot \min  _ {c _ {1}, \dots , c _ {k} \in \mathbb {R} ^ {t}} \sum_ {x \in X} \min  _ {j \in [ k ]} \| \Pi (x) - c _ {j} \| _ {2} ^ {z}.
$$
$$
\operatorname {c o s t} _ {z} \left(Y _ {1}, \dots , Y _ {k}\right) \leq \left(2 ^ {z - 1} + 2 ^ {3 z - 2} \mathrm {D} _ {1} ^ {z} \mathrm {D} _ {2} (1 + \mathrm {D} _ {3})\right) \cdot \operatorname {c o s t} _ {z} \left(X _ {1} ^ {*}, \dots , X _ {k} ^ {*}\right).
$$
Before starting the proof of Lemma B.3, we show that projecting points onto a random Gaussian vector gives the first two desired guarantees of the above lemma with  $\mathsf{D}_1 \coloneqq (k^2/\delta)$  and  $\mathsf{D}_2 \coloneqq 2^{O(z)}/\delta$  with probability at least  $1 - \delta$ . The first lemma that we state below shows that the first condition of Lemma B.3 is satisfied with high probability, and the second lemma that the second condition of Lemma B.3 is satisfied with high probability.
Lemma B.4 (Centers Don't Contract). Let  $C = \{c_1, \ldots, c_k\} \subset \mathbb{R}^d$  denote any collection of  $k$  points and let  $\Pi \colon \mathbb{R}^d \to \mathbb{R}$  be a random map given by
$$
\boldsymbol {\Pi} (x) = \langle x, \mathbf {g} \rangle
$$
for a randomly chosen vector  $\mathbf{g} \sim \mathcal{N}(0, I_d)$ . Then, with probability at least  $1 - \delta$  over  $\mathbf{g}$ , every  $i, j \in [k]$  satisfies
$$
\left(\frac {\delta}{k ^ {2}}\right) \cdot \| c _ {i} - c _ {j} \| _ {2} \leq \| \boldsymbol {\Pi} (c _ {i}) - \boldsymbol {\Pi} (c _ {j}) \| _ {2}.
$$
Lemma B.5 (Cost of  $(X_1^*,\ldots ,X_k^*)$  does not Increase). Let  $X = \{x_{1},\dots,x_{n}\} \subset \mathbb{R}^{d}$  and let  $X_{1}^{*},\ldots ,X_{k}^{*}$  be the partition of  $X$ , and  $c_{1}^{*},\ldots ,c_{k}^{*}\in \mathbb{R}^{d}$  be the centers which minimize the  $(k,z)$ -clustering cost of  $X$ . Then, with probability a least  $1 - \delta$ ,
$$
\sum_ {i = 1} ^ {k} \sum_ {x \in X _ {i} ^ {*}} \| \boldsymbol {\Pi} (x) - \boldsymbol {\Pi} \left(c _ {i} ^ {*}\right) \| _ {2} ^ {z} \leq \left(2 ^ {O (z)} / \delta\right) \sum_ {i = 1} ^ {k} \sum_ {x \in X _ {i} ^ {*}} \| x - c _ {i} ^ {*} \| _ {2} ^ {z}.
$$
Proof of Lemma B.1 assuming Lemma B.3, Lemma B.4 and Lemma B.5. We will apply Lemma B.3 by letting  $\delta$  be a small enough constant (say,  $\delta = 0.01$ ) to take a union bound. Lemma B.4 implies the first condition with  $\mathsf{D}_1 = O(k^2)$  and Lemma B.5 implies the second condition with  $\mathsf{D}_2 = 2^{O(z)}$ . Finally, the second assumption of Lemma B.1 sets  $r = \mathsf{D}_3$ , from which we derive the conclusion.
We now prove Lemma B.4, Lemma B.5. Lemma B.3 is proved in Subsection B.2.
Proof of Lemma B.4. The proof relies on the 2-stability property of the Gaussian distribution. Namely, if we let  $z \in \mathbb{R}^d$  be an arbitrary vector and we sample a standard Gaussian vector  $\mathbf{g} \sim \mathcal{N}(0, I_d)$ , the (scalar) random variable  $\langle z, \mathbf{g} \rangle$  is distributed like  $\|z\|_2 \cdot \mathbf{g}'$ , where  $\mathbf{g} \sim \mathcal{N}(0, 1)$ . Using the 2-stability of the Gaussian distribution for every  $i, j \in [k]$ , we have that  $\|\Pi(c_i) - \Pi(c_j)\|_2^2$  is distributed as  $(\mathbf{g}')^2 \|c_i - c_j\|_2^2$ , where  $\mathbf{g}'$  is distributed as a (one-dimensional) Gaussian  $\mathcal{N}(0, 1)$ . Thus, by a union bound, the probability that there exists a pair  $i, j \in [k]$ , which satisfies  $\|\Pi(c_i) - \Pi(c_j)\|_2^2 < \alpha^2 \cdot \|c_i - c_j\|_2^2$  is at most  $k^2$  times the probability that a Gaussian random variable lies in  $[- \alpha, \alpha]$ , and this probability is easily seen to be less than  $\alpha$ . Setting  $\alpha = \delta / k^2$  gives the desired lemma.
Proof of Lemma B.5. Similarly to the proof of Lemma B.4, we have that  $\| \Pi(x) - \Pi(c_i^*) \|_2^z$  is distributed as  $|g'|^z \cdot \| x - c_i^* \|_2^z$ , where  $g'$  is distributed as a (one-dimensional) Gaussian  $\mathcal{N}(0,1)$ . By linearity of expectation,
$$
\mathbb {E} _ {\boldsymbol {\Pi} \sim \mathcal {I} _ {d}} \left[ \sum_ {i = 1} ^ {k} \sum_ {x \in X _ {i} ^ {*}} \| \boldsymbol {\Pi} (x) - \boldsymbol {\Pi} (c _ {i} ^ {*}) \| _ {2} ^ {z} \right] = \sum_ {i = 1} ^ {k} \sum_ {x \in X _ {i} ^ {*}} \underset {\mathbf {g} ^ {\prime} \sim \mathcal {N} (0, 1)} {\mathbb {E}} [ | \mathbf {g} _ {i} ^ {\prime} | ^ {z} ] \cdot \| x - c _ {i} ^ {*} \| _ {2} ^ {z}.
$$
To conclude, note that for  $z \geq 1$ , there is some  $\alpha > 1$  such that  $\alpha z$  is an even integer and  $\alpha \leq 2$ . Thus, by Jensen's inequality and the fact that  $f(x) = x^{1 / \alpha}$  is concave we can write
$$
\underset {\mathbf {g} ^ {\prime} \sim \mathcal {N} (0, 1)} {\mathbb {E}} \left[ | \mathbf {g} ^ {\prime} | ^ {z} \right] \leq \left(\mathbb {E} \left[ (\mathbf {g} ^ {\prime}) ^ {\alpha z} \right]\right) ^ {1 / \alpha}
$$
Now note that all odd moments of the Gaussian distribution are zero by symmetry. Thus, for the moment generating function  $\mathbb{E}[e^{\mathbf{g}'}]$  it holds that
$$
\mathbb {E} \left[ e ^ {\mathbf {g} ^ {\prime}} \right] = \sum_ {k = 0} ^ {\infty} \frac {1}{(2 k) !} \cdot \mathbb {E} \left[ \left(\mathbf {g} ^ {\prime}\right) ^ {2 k} \right].
$$
As  $\mathbb{E}[e^{\mathbf{g}'}]\leq e^{1 / 2}$  it follows that
$$
\left(\mathbb {E} \left[ \left(\mathbf {g} ^ {\prime}\right) ^ {\alpha z} \right]\right) ^ {1 / \alpha} \leq \left((\alpha z)! \mathbb {E} \left[ e ^ {\mathbf {g} ^ {\prime}} \right]\right) ^ {1 / \alpha} \leq \left((\alpha z)! e ^ {1 / 2}\right) ^ {1 / \alpha} \leq 2 ^ {O (z)}.
$$
Applying Markov's inequality now completes the proof.
B.2 Proof of Lemma B.3
Let  $\{\hat{c}_i\}_{i\in [k]}$  be an optimal set of centers for the partition  $(Y_{1},\ldots ,Y_{k})$  for the  $(k,z)$ -clustering problem on  $\Pi (X)$ , where  $\hat{c}_i\in Y_i$ . Specifically, the points  $\hat{c}_1,\dots ,\hat{c}_k\in \mathbb{R}^t$  are those which minimize
$$
\operatorname {c o s t} \left(\Pi \left(Y _ {1}\right), \dots , \Pi \left(Y _ {k}\right)\right) \stackrel {\text {d e f}} {=} \sum_ {i = 1} ^ {k} \sum_ {x \in Y _ {i}} \| \Pi (x) - \hat {c} _ {i} \| _ {2} ^ {z}.
$$
To quantize the cost difference between the centers  $c^*$  and the centers  $\hat{c}$  we analyze the following value. We assume we mapped every point of  $X$  to  $\Pi c^*(X)$ , and we then compute the cost of the partition  $Y_1, \ldots, Y_k$  on this set. Formally, we let
$$
\operatorname {v a l} (\Pi , c ^ {*}, Y _ {1}, \dots , Y _ {k}) = \sum_ {i = 1} ^ {k} \sum_ {j = 1} ^ {k} | Y _ {i} \cap X _ {j} ^ {*} | \cdot \| \Pi (c _ {j} ^ {*}) - \hat {c} _ {i} \| _ {2} ^ {z}.
$$
First, we prove the following simple claim.
Claim B.6. There exists a set of centers  $c_1', \ldots, c_k'$  (with possible repetitions) which are chosen among the points  $\{c_1^*, \ldots, c_k^*\}$  such that
$$
\sum_ {i = 1} ^ {k} \sum_ {j = 1} ^ {k} \left| Y _ {i} \cap X _ {j} ^ {*} \right| \cdot \left\| \Pi \left(c _ {j} ^ {*}\right) - \Pi \left(c _ {i} ^ {\prime}\right) \right\| _ {2} ^ {z} \leq 2 ^ {z} \cdot \operatorname {v a l} \left(\Pi , c ^ {*}, Y _ {1}, \dots , Y _ {k}\right).
$$
Proof. We will prove the claim using the probabilistic method. For every  $i \in [k]$ , consider the distribution over center  $\{c_1^*, \ldots, c_k^*\}$  which samples a center  $c_i'$  as
$$
\operatorname * {P r} _ {\boldsymbol {c} _ {i} ^ {\prime}} \left[ \boldsymbol {c} _ {i} ^ {\prime} = c _ {j} ^ {*} \right] = \frac {| Y _ {i} \cap X _ {j} ^ {*} |}{\sum_ {\ell = 1} ^ {d} | Y _ {i} \cap X _ {j} ^ {*} |}.
$$
Then, we upper bound the expected cost of using the centers  $c_i'$ . Using Lemma A.3,
$$
\begin{array}{l} \mathbb {E} \left[ \sum_ {i = 1} ^ {k} \sum_ {j = 1} ^ {k} \left| Y _ {i} \cap X _ {j} ^ {*} \right| \cdot \left\| \Pi \left(c _ {j} ^ {*}\right) - \Pi \left(\boldsymbol {c} _ {i} ^ {\prime}\right) \right\| _ {2} ^ {z} \right] \\ \leq \sum_ {i = 1} ^ {k} \sum_ {j = 1} ^ {k} \left| Y _ {i} \cap X _ {j} ^ {*} \right| \cdot \mathbb {E} \left[ \left(\left\| \Pi \left(c _ {j} ^ {*}\right) - \hat {c} _ {i} \right\| _ {2} + \left\| \Pi \left(\boldsymbol {c} _ {i} ^ {\prime}\right) - \hat {c} _ {i} \right\| _ {2}\right) ^ {z} \right] \\ \leq 2 ^ {z - 1} \sum_ {i = 1} ^ {k} \sum_ {j = 1} ^ {k} | Y _ {i} \cap X _ {j} ^ {*} | \cdot \| \Pi \left(c _ {j} ^ {*}\right) - \hat {c} _ {i} \| _ {2} ^ {\tilde {z}} + 2 ^ {z - 1} \sum_ {i = 1} ^ {k} \left(\sum_ {j = 1} ^ {k} | Y _ {i} \cap X _ {j} ^ {*} |\right) \mathbb {E} \left[ \| \Pi \left(\boldsymbol {c} _ {i} ^ {\prime}\right) - \hat {c} _ {i} \| _ {2} ^ {\tilde {z}} \right] \\ = 2 ^ {z - 1} \cdot \operatorname {v a l} \left(\Pi , c ^ {*}, Y _ {1}, \dots , Y _ {k}\right) + 2 ^ {z - 1} \sum_ {i = 1} ^ {k} \left(\sum_ {j = 1} ^ {k} \left| Y _ {i} \cap X _ {j} ^ {*} \right|\right) \sum_ {\ell = 1} ^ {k} \frac {\left| Y _ {i} \cap X _ {\ell} ^ {*} \right|}{\sum_ {j = 1} ^ {k} \left| Y _ {i} \cap X _ {j} ^ {*} \right|} \cdot \| \Pi \left(c _ {\ell} ^ {*}\right) - \hat {c} _ {i} \| _ {2} ^ {z} \\ = 2 ^ {z} \cdot \operatorname {v a l} \left(\Pi , c ^ {*}, Y _ {1}, \dots , Y _ {k}\right). \\ \end{array}
$$