SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Theorem 2.3. For any input  $\mathbf{A} \in \mathbb{R}^{n \times d}$  and  $\epsilon \in (0,1)$ , Algorithm [2] will return a feasible solution to the  $k$ -means clustering problem (Definition [1.2],  $(\tilde{\mathbf{X}},\tilde{\mathbf{D}})$ , satisfying:
$$
\| \tilde {\mathbf {X}} \tilde {\mathbf {D}} - \mathbf {A} \| _ {F} \leq (1 + \epsilon) \cdot \min  _ {\mathbf {D} \in \mathbb {R} ^ {k \times d}, \mathbf {X} \in \mathcal {X}} \| \mathbf {X} \mathbf {D} - \mathbf {A} \| _ {F},
$$
with constant probability. Furthermore, Algorithm 2 runs in  $n \cdot \mathrm{poly}(k / \epsilon) + \exp \left( \frac{k}{\epsilon} \right)$  polylog(k/ε)) time.
3 Turnstile streaming algorithms
In this section, we consider the turnstile streaming model (see Definition 1.3). We provide upper bounds on the space needed to compute an  $\epsilon$ -relative error solution to the  $k$ -means problem and a restricted form of the sparse dictionary learning problem in a turnstile stream. We do this by showing that these approximately optimal solutions can be computed from a few small linear sketches of the original data matrix, and any linear sketch can be trivially maintained in a turnstile stream by linearity of the updates. A key idea behind these algorithms is applying the guess-the-sketch approach introduced in Razenshteyn et al. [2016] along with the following theorem.
Theorem 3.1. (Theorem 3.1 in Clarkson and Woodruff [2009]) Given  $\delta, \epsilon > 0$ , suppose  $\mathbf{A}$  and  $\mathbf{B}$  are matrices with  $n$  rows, and  $\mathbf{A}$  has rank at most  $k$ . There is an  $m = O(k \log(1/\delta)/\epsilon)$  such that, if  $\mathbf{S}$  is an  $m \times n$  sign matrix, then with probability at least  $1 - \delta$ , if  $\tilde{\mathbf{X}} = \mathrm{argmin}_{\mathbf{X}} \|\mathbf{S}(\mathbf{AX} - \mathbf{B})\|_F^2$  and  $\mathbf{X}^* = \mathrm{argmin}_{\mathbf{X}} \|\mathbf{AX} - \mathbf{B}\|_F^2$ , then  $\|\mathbf{A}\tilde{\mathbf{X}} - \mathbf{B}\|_F \leq (1 + \epsilon) \|\mathbf{AX}^* - \mathbf{B}\|_F$ .
Notice that, if we knew the optimal solution  $\mathbf{X}^*$  exactly, then by the previous theorem we could compute an approximately optimal dictionary  $\tilde{\mathbf{D}}$  exactly as  $\tilde{\mathbf{D}} = (\mathbf{S}\mathbf{X}^{*})^{\dagger}\mathbf{S}\mathbf{A}$ . The key observation is that, since  $\mathbf{S}$  is a random sign matrix and the rows of  $\mathbf{X}$  are standard basis vectors, the set  $\{\mathbf{S}\mathbf{X} \mid \mathbf{X} \in \mathcal{X}, \mathbf{S} \in \{\pm 1\}^{\tilde{\mathcal{O}}(k / \epsilon) \times n}\}$  is not too large. Also, we can approximately solve  $\min_{\mathbf{X} \in \mathcal{X}} \| \mathbf{X}\tilde{\mathbf{D}} - \mathbf{A}\|_F^2$  for a fixed  $\tilde{\mathbf{D}}$  with constant probability by solving  $\tilde{\mathbf{X}} = \min_{\mathbf{X} \in \mathcal{X}} \| (\mathbf{X}\tilde{\mathbf{D}} - \mathbf{A})\mathbf{T}\|_F^2$ , where  $\mathbf{T}$  is a moderately sized affine embedding matrix. Since the number of possible  $(\tilde{\mathbf{X}},\tilde{\mathbf{D}})$  is not too large, an  $\ell_2$ -embedding matrix,  $\mathbf{W}$ , can be used to approximate  $\|\tilde{\mathbf{X}}\tilde{\mathbf{D}} - \mathbf{A}\|_F^2$  for every possible  $(\tilde{\mathbf{X}},\tilde{\mathbf{D}})$ .
Our streaming algorithm relies on carefully balancing the roles of the three sketching matrices to minimize the size of the sketches, using the weakest guarantee possible for each component. In particular, it is critical to use the affine embedding matrix  $\mathbf{T}$  to only preserve the error for a fixed  $\tilde{\mathbf{D}}$  instead of every subproblem and instead use the  $\ell_2$ -embedding matrix  $\mathbf{W}$  to identify which subproblem provides an approximate solution to the overall problem.
Theorem 3.2. (1) There are distributions of random sketching matrices  $\mathbf{T} \in \mathbb{R}^{d \times t}$ ,  $\mathbf{S} \in \mathbb{R}^{s \times n}$ , and  $\mathbf{W} \in \mathbb{R}^{w \times nd}$ , with  $t = \mathcal{O}(\log(nk) / \epsilon^2)$ ,  $s = \mathcal{O}\left(\frac{k}{\epsilon}\right)$ , and  $w = \mathcal{O}\left(\frac{k^2}{\epsilon^3} \log(n)\right)$  such that SA, AT, and W vec(A) suffice to compute a  $(1 + \epsilon)$ -approximate solution to the  $k$ -means problem with at least constant probability, where vec(A)  $\in \mathbb{R}^{nd}$  is the flattening of A.
(2) There is an algorithm which computes a  $(1 + \epsilon)$ -approximate solution to the  $k$ -means problem in the turnstile model with at least constant probability using  $\tilde{\mathcal{O}}(n/\epsilon^2 + dk/\epsilon)$  space for  $n, d > \mathrm{poly}(k/\epsilon)$  in  $n^{\tilde{\mathcal{O}}(k^2/\epsilon)}$  additional time.
The previous proof critically relies on the fact that  $\{\mathbf{S}\mathbf{X} \mid \mathbf{X} \in \mathcal{X}, \mathbf{S} \in \{\pm 1\}^{m \times n}\}$  is a finite set that is not too large. We must therefore introduce the following restricted form of the sparse dictionary problem.
Definition 3.1. (Discrete  $r$ -sparse dictionary problem) Let  $\mathcal{X}$  be the space of  $n \times k$  matrices with at most  $r$  non-zero entries per row and non-zero entries taking values in  $\{-D, -(D - 1), \dots, -1, 0, 1, \dots (D - 1), D\}$ . The goal of this problem is to solve the following optimization problem:
$$
\mathbf{X}^{*},\mathbf{D}^{*} = \operatorname *{argmin}_{\mathbf{X}\in \mathcal{X},\mathbf{D}\in \mathbb{R}^{k\times d}}\| \mathbf{XD} - \mathbf{A}\|_{F},
$$
where  $\mathbf{A}\in \mathbb{R}^{n\times d}$  is an arbitrary input matrix.
Under this constraint that the solution is in a discrete space the proof of the streaming algorithm for sparse dictionary learning proceeds essentially the same as for  $k$ -means while accounting for the larger solution space.
Theorem 3.3. (1) There are distributions of random sketching matrices  $\mathbf{T} \in \mathbb{R}^{d \times t}$ ,  $\mathbf{S} \in \mathbb{R}^{s \times n}$ , and  $\mathbf{W} \in \mathbb{R}^{w \times nd}$ , with  $t = \mathcal{O}(r\log (nkD) / \epsilon^2)$ ,  $s = \mathcal{O}\left(\frac{k}{\epsilon}\right)$ , and  $w = \mathcal{O}\left(\frac{k^2}{\epsilon^3}\log (nd)\right)$  such that SA, AT, and W vec(A) suffice to compute a  $(1 + \epsilon)$ -approximate solution to the discrete r-sparse dictionary problem (Definition 3.1) with at least constant probability.
(2) There is an algorithm which computes a  $(1 + \epsilon)$ -approximate solution to the  $r$ -sparse dictionary problem in the turnstile model with at least constant probability using  $\tilde{\mathcal{O}}(nr / \epsilon^2 + dk / \epsilon)$  space for  $n, d > \mathrm{poly}(k / \epsilon)$  in  $k^r \cdot (nD)^{\tilde{\mathcal{O}}(k^2 / \epsilon)}$  additional time.
Removing the restriction that  $\mathbf{X}^*$  belongs to the restricted space would be an interesting future problem. However, two issues are that the entries of  $\mathbf{X}$  may be very large, since the rows of  $\mathbf{D}$  may not be orthogonal, and a uniform discretization is required to apply a guess-the-sketch argument.
4 Streaming lower bounds for Euclidean  $k$ -means clustering
We introduce slightly different definitions of the  $k$ -means clustering problem than the one used in Definition 1.2 to facilitate the notation of our lower bound arguments in this section.
Definition 4.1 ( $k$ -means clustering cost). Let  $\{a^i\}_{i=1}^n \subseteq \mathbb{R}^d$  be a set of  $n$  vectors in  $d$  dimensions. Then, we define the  $k$ -means clustering cost of centers  $c^1, c^2, \ldots, c^k \in \mathbb{R}^d$  to be
$$
\operatorname {c o s t} \left(c ^ {1}, c ^ {2}, \dots , c ^ {k}\right) := \sum_ {i = 1} ^ {n} \min  _ {j = 1} ^ {k} \| a ^ {i} - c ^ {j} \| _ {2} ^ {2}.
$$
Definition 4.2 (Approximate solutions to  $k$ -means clustering). Let  $\{a^i\}_{i=1}^n \subseteq \mathbb{R}^d$  be a set of  $n$  vectors in  $d$  dimensions. Let
$$
\mathsf {O P T} := \min  _ {c ^ {1}, c ^ {2}, \dots , c ^ {k} \in \mathbb {R} ^ {d}} \operatorname {c o s t} (c ^ {1}, c ^ {2}, \dots , c ^ {k})
$$
We say that an algorithm outputs an  $\epsilon$ -approximate solution to the  $k$ -means clustering problem if the algorithm outputs one of the following:
- Partition: a partition  $C^1, C^2, \ldots, C^k \subseteq [n]$  such that
$$
\sum_ {j = 1} ^ {k} \sum_ {i \in C ^ {j}} \| a ^ {i} - \hat {c} ^ {j} \| _ {2} ^ {2} \leq (1 + \epsilon) \mathsf {O P T}
$$
where  $\hat{c}^j \coloneqq \frac{1}{|C^j|}\sum_{i\in C^j}a^i$
- Centers: centers  $\hat{c}^1, \hat{c}^2, \dots, \hat{c}^k \in \mathbb{R}^d$  such that  $\mathrm{cost}(\hat{c}^1, \hat{c}^2, \dots, \hat{c}^k) \leq (1 + \epsilon)\mathrm{OPT}$ .
- Cost: a number  $c \geq 0$  such that  $\mathsf{OPT} \leq c \leq (1 + \epsilon) \mathsf{OPT}$ .
4.1 Lower bounds for  $k$ -means clustering
Our most technically involved and delicate lower bound result is the following theorem, which shows that nearly optimally solving  $k$ -means clustering to  $(1 + \epsilon)$  accuracy requires  $\tilde{\Omega}(n / \epsilon)$  bits of space:
Theorem 1.1 (Informal restatement of Theorem C.1). Let  $k = d = \tilde{O}(1/\epsilon)$ . Suppose a turnstile streaming algorithm outputs centers  $\{\hat{c}^j\}_{j=1}^k \subseteq \mathbb{R}^d$  as well as assignments of  $n$  points to the  $k$  centers, which achieves a  $(1+\epsilon)$ -approximately optimal solution to the  $k$ -means clustering problem. Then, the algorithm must use at least  $\tilde{\Omega}(n/\epsilon)$  bits of space over any constant number of passes.
We defer the full proof to Appendix C and give a proof sketch in this section to illustrate the most important ideas.
The hard instance: set disjointness. The starting point to our lower bound is the information theoretic communication complexity lower bound for the set disjointness problem due to Bar-Yossef et al. [2004]. In the two-party set disjointness problem, two players Alice and Bob each have a bit vector  $A, B \in \{0,1\}^d$  in  $d$  dimensions, and they must determine whether there exists a coordinate  $j \in [d]$  such that  $A_j = B_j = 1$  or not. The work of Bar-Yossef et al. [2004] shows that in order to solve this problem, Alice and Bob must exchange messages that reveal at least  $\Omega(d)$  bits of information about their inputs, which in turn implies an  $\Omega(d)$  communication complexity lower bound for this problem, as well as an  $\Omega(nd)$  communication complexity lower bound for solving a constant fraction of  $n$  independent instances of the same problem. Furthermore, the hard instance of Bar-Yossef et al. [2004] has a simple input distribution: the vectors  $(A, B)$  are such that the  $j$ th coordinate  $(A^j, B^j)$  is drawn either as  $(0,0)$  with probability  $1/2$  or  $(1,0)$  with probability  $1/4$  or  $(0,1)$  with probability  $1/4$ , except for one coordinate, which may take the value  $(1,1)$ .
We aim to make use of this result as follows. Consider the vector  $Z = A + B$ . This vector has entries in  $\{0,1\}$ , except possibly for one entry, which could be 2. If we have  $n$  such vectors, then we expect a good clustering into  $k = d$  clusters to cluster all points with  $Z_{j} = 2$  together. Such a clustering would be able to output the index of the intersection of  $A$  and  $B$ , which intuitively requires more information than just determining whether there is an intersection or not, and thus should also require  $\Omega(d)$  bits of information cost. Furthermore, we can choose the dimension  $d$  to be roughly  $1 / \epsilon$ , so that the cost of clustering  $Z$  to the "correct" center will have a cost of  $\Theta(d) = \Theta(1 / \epsilon)$ , while clustering  $Z$  to the incorrect center will incur an additional error of  $\Theta(1)$ , which is an  $\epsilon$  fraction of the cost.
Cost calculations. The main challenge in carrying out the idea in the previous paragraph is in arguing that the target optimal clustering that we wish to discover indeed is a nearly optimal clustering, and that significant deviations from this clustering result in a large cost. This involves showing a lower bound on the cost of any clustering.
Our first step is to obtain a lower bound on the cost of any clustering of  $n$  random bit vectors in  $d$  dimensions. If we first fix a set of  $k$  centers  $\{c^j\}_{j=1}^k$ , then the minimum distance between a random bit vector  $Z$  and any of the  $c^j$  can be bounded by using Chernoff bounds, which implies a lower bound of  $d/4 - O(\log d)$  on this quantity in expectation (Lemma C.4). Note, however, that this lower bound is not high enough to prevent a nearly optimal solution from just assigning points according to the best clustering of the random bits while ignoring the one entry that takes the value of  $Z_j = 2$ , which means that the clustering need not solve the problem of identifying the intersection coordinate between  $A$  and  $B$ .
To address this problem, we need to make the cost of ignoring the intersection coordinate much more costly. We do this by instead considering the multi-party set disjointness problem, so that we now have  $t = O(\sqrt{\log d})$  players rather than just 2, each with an input vector  $A^{(i)} \in \{0,1\}^d$ , so that  $Z = \sum_{i=1}^{t} A^{(i)}$  is now a random bit vector except for a single entry with a  $t$  rather than a 2. Now, a clustering which does not correctly identify the intersection coordinate will pay a cost of roughly  $t^2 = O(\log d)$ , which is large enough to overcome the potential savings from a good clustering of the random bit coordinates. We also "plant" the target centers  $c^j$  by adding roughly  $n/k$  copies of each of our target centers  $c^j$  as part of the input instance (Lemma C.7), so that choosing centers  $\hat{c}^j$  that are significantly different from  $c^j$  must incur a large cost. In particular, we can get the guarantee that on average,  $\| c^j - \hat{c}^j \|_2^2 \leq o(1)$ .
At this point, we can argue that most of the  $k$  centers are the centers that expect, i.e., roughly  $t$  on one coordinate and  $1/2$  on the rest of the coordinates. Thus, if we cluster a point  $Z$  whose center we expect to be  $\hat{c}^j$  but is clustered to some other  $\hat{c}^{j'}$ , and furthermore  $\hat{c}^{j'}$  is close to our expected center  $c^{j'}$ , then we must incur an additional  $O(\log d)$  cost which is too expensive. However, there is still the possibility that for the very small number of clusters  $\hat{c}^j$  which do not satisfy  $\| c^j - \hat{c}^j \|_2^2 \leq o(1)$ , these centers could be assigned a very large number of points with very low cost. We also show that this cannot be the case, by arguing that if a large number of points are assigned to very few clusters, then the cost must be large (Lemma C.8). With this lemma in hand, we are able to show our main result in Theorem C.1 by carefully combining the various cost contribution bounds discussed previously.
4.1.1 Lower bound for outputting nearly optimal centers
We note that an  $\Omega (dk / \epsilon)$  lower bound follows from an earlier lower bound for low rank approximation due to Woodruff [2014a], even for row arrival streams:
Definition 4.3 (Row arrival stream). We say that an algorithm outputs an  $\epsilon$ -approximate solution to the  $k$ -means clustering problem in the row arrival streaming model if the input vectors  $\{a^i\}_{i=1}^n \subseteq \mathbb{R}^d$  arrive one at a time.
Theorem 4.1. Suppose that an algorithm outputs centers  $\{\hat{c}^j\}_{j=1}^k \subseteq \mathbb{R}^d$  that achieves a  $(1 + \epsilon)$ -approximately optimal solution to the  $k$ -means clustering problem after one pass through a row arrival stream (Definition 4.3). Then, the algorithm must use at least  $\tilde{\Omega}(dk/\epsilon)$  bits of space.
We briefly justify why the techniques of Woodruff [2014a] imply Theorem 4.1. The result of Woodruff [2014a] constructs a distribution over  $O(k / \epsilon) \times d$  matrices such that one can recover an arbitrary random bit among  $\tilde{\Omega}(dk / \epsilon)$  random bits by appending a set of  $k$  "query" rows and then computing a  $(1 + \epsilon)$ -approximately optimal low rank approximation to the resulting matrix. Furthermore, it is shown that a nearly optimal rank  $k$  approximation is obtained by approximating all but  $k$  rows by zero vectors. Such a rank  $k$  approximation in fact corresponds to a clustering solution, and thus the proof of Woodruff [2014a] immediately applies to our  $k$ -means clustering setting as well.
4.2 Lower bounds for center cost query data structures
Next, we study lower bounds against streaming algorithms which have the guarantee of approximating the cost of an arbitrary but fixed set of centers. We formalize the guarantee we study in Definition 4.4.
Definition 4.4 (Center cost query data structure). We say that  $\mathcal{Q}$  is an  $\epsilon$ -approximate center cost query data structure for the  $k$  means clustering problem for the instance  $\{a^i\}_{i=1}^n$  if, for any centers  $c^1, c^2, \ldots, c^k \in \mathbb{R}^d$ ,  $\mathcal{Q}$  outputs one of the following:
$$
\sum_ {j = 1} ^ {k} \sum_ {i \in C ^ {j}} \| a ^ {i} - c ^ {j} \| _ {2} ^ {2} \leq (1 + \epsilon) \operatorname {c o s t} (c ^ {1}, c ^ {2}, \dots , c ^ {k}).
$$
- Cost: a number  $c \geq 0$  such that
$$
\operatorname {c o s t} \left(c ^ {1}, c ^ {2}, \dots , c ^ {k}\right) \leq c \leq (1 + \epsilon) \operatorname {c o s t} \left(c ^ {1}, c ^ {2}, \dots , c ^ {k}\right)
$$
Our first lower bound is an  $\Omega(n/\epsilon^2)$  bit space lower bound for a center cost query data structure which can output a partition for  $k$ -means clustering with  $k = 2$ . We proceed by a standard encoding argument, showing that any such data structure must encode  $\Omega(n/\epsilon^2)$  many random bits. We provide the full proof in Appendix D.1.
Theorem 4.2. Let  $\epsilon \in (0,1/3)$  and  $k = 2$ . Suppose that an algorithm maintains an  $\epsilon/15$ -approximate center cost query data structure for  $k$ -means clustering that outputs a partition (Definition 4.4) over a row arrival stream (Definition 4.3). Then, the algorithm must use at least  $\Omega(n/\epsilon^2)$  bits of space, over any constant number of passes.
4.3 Approximation of costs and centers
We show  $\Omega (n)$  space memory bounds when we only need to estimate the optimal cost or centers achieving nearly optimal cost, up to a constant factor. Our lower bounds in this section are simpler reductions from the set disjointness problem Razborov [1990], Bar-Yossef et al. [2004]. Proofs are provided in Appendix D.2 and D.3.
Theorem 4.3 (Lower Bound for Estimating  $k$ -Means Clustering Cost). Let  $k = 2$  and let  $\mathcal{X}$  be the set of matrices  $\mathbf{X} \in \mathbb{R}^{n \times k}$  with standard basis vectors as rows. Let  $d = 1$ . Any randomized algorithm which outputs a number  $c \geq 0$  satisfying
$$
c \leq \min  _ {\mathbf {X} \in \mathcal {X}, \mathbf {D} \in \mathbb {R} ^ {k \times d}} \| \mathbf {X D} - \mathbf {A} \| _ {F} ^ {2} <   2 c \tag {2}
$$
in a constant number of passes over a turnstile stream requires  $\Omega(n)$  bits of space.
Theorem 4.4 (Lower Bound for Computing Approximate Centers). Let  $k = 3$  and let  $\mathcal{X}$  be the set of matrices  $\mathbf{X} \in \mathbb{R}^{n \times k}$  with standard basis vectors as rows. Let  $d = 1$ . Any randomized algorithm which outputs centers  $\tilde{\mathbf{D}} \in \mathbb{R}^{k \times d}$  satisfying
$$
\min  _ {\mathbf {X} \in \mathcal {X}} \| \mathbf {X} \tilde {\mathbf {D}} - \mathbf {A} \| _ {F} ^ {2} <   2 \min  _ {\mathbf {X} \in \mathcal {X}, \mathbf {D} \in \mathbb {R} ^ {k \times d}} \| \mathbf {X} \mathbf {D} - \mathbf {A} \| _ {F} ^ {2}
$$
in a constant number passes over a turnstile stream requires  $\Omega(n)$  bits of space.
4.4 New upper bounds in random order streams
In this section, we show some new upper bounds showing that we can go beyond the previously presented lower bounds. In particular, in random order row arrival streams with bounded sensitivity, we show that the first segment of the stream is sufficient to obtain approximately optimal centers, and these can in turn be used to nearly optimally cluster the rest of the stream. We give the full proof of this result in Appendix D.4
Theorem 4.5. Suppose that the rows of  $\mathbf{A} \in \mathbb{R}^{n \times d}$  arrive in a random order row arrival stream. Furthermore, suppose that the sensitivities of each row  $a^i$  are bounded by  $\alpha$ , that is,
$$
\sup  _ {c ^ {1}, c ^ {2}, \dots , c ^ {k} \in \mathbb {R} ^ {d}} \frac {\min  _ {j = 1} ^ {k} \| a ^ {i} - c ^ {j} \| _ {2} ^ {2}}{\sum_ {i ^ {\prime} = 1} ^ {n} \min  _ {j = 1} ^ {k} \| a ^ {i ^ {\prime}} - c ^ {j} \| _ {2} ^ {2}} \leq \alpha .
$$
Then, there is an algorithm which, with constant probability, outputs a  $(1 + \epsilon)$ -nearly optimal clustering with partitions and centers using
$$
\tilde {O} \left(\alpha n k d / \epsilon^ {4} + d k / \epsilon + n\right).
$$
bits of space. In particular, if  $\alpha \leq \epsilon^4 /kd$ , then this algorithm uses just  $\tilde{O} (n + dk / \epsilon)$  bits of space.
5 Open directions
We conclude with several questions left open by our work.
1. In our PTAS for sparse dictionary learning of Theorem 2.2, can the bit complexity assumption be removed?
2. In the turnstile streaming setting, our main question is settling the space complexity of  $k$ -means clustering with assignments. Currently, the upper bound is  $\tilde{O}(n/\epsilon^2)$  bits whereas our lower bound in Theorem C.1 is  $\tilde{\Omega}(n/\epsilon)$  bits. Can this  $\epsilon$  factor gap be closed by improving the upper bound or the lower bound?
3. In random order streaming model, we gave an  $k$ -means clustering upper bound using a bounded sensitivity assumption in Theorem 4.5. Can this assumption be removed? What upper bounds and lower bound are possible in this model?
We thank the anonymous reviewers for useful feedback on improving the presentation of this work. Petros Drineas and Gregory Dexter were partially supported by NSF AF 1814041, NSF FRG 1760353, and DOE-SC0022085. David P. Woodruff and Taisuke Yasuda were supported by a Simons Investigator Award.
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