SlowGuess/NeurIPS_2023_OCR huggingface dataset:
[IK22] gave the first algorithm for general list-decodable covariance estimation that achieves nontrivial bounds in total variation distance using the powerful sum-of-squares hierarchy. Their algorithm outputs an  $\exp (\mathrm{poly}(1 / \alpha))$ -sized list of matrices containing an  $H_{i}$  that is close to the true  $\Sigma$  in two metrics (i) relative Frobenius norm:  $\| H_i^{-1 / 2}\Sigma H_i^{-1 / 2} - I\| _\mathrm{F} = \mathrm{poly}(1 / \alpha)$  and (ii) multiplicative spectral approximation:  $\mathrm{poly}(\alpha)\Sigma \preceq H_i\preceq \mathrm{poly}(1 / \alpha)\Sigma$ . Their algorithm uses  $d^{\mathrm{poly}(1 / \alpha)}$  samples, which seems to be necessary for efficient (statistical query) algorithms achieving multiplicative spectral approximation [DKS17; DKPPS21]. In comparison, Theorem 1.2 uses only  $\mathrm{poly}(d / \alpha)$  samples, returns a list of  $O(1 / \alpha)$  matrices, but approximates only in the relative Frobenius norm:  $\| H_i^{-1 / 2}\Sigma H_i^{-1 / 2} - I\| _\mathrm{F} = \mathrm{poly}(1 / \alpha)$ .
For a vector  $v$ , we let  $\| v \|_2$  denote its  $\ell_2$ -norm. We use  $I_d$  to denote the  $d \times d$  identity matrix; We will drop the subscript when it is clear from the context. For a matrix  $A$ , we use  $\| A \|_{\mathrm{F}}$  and  $\| A \|_{\mathrm{op}}$  to denote the Frobenius and spectral (or operator) norms, respectively. We denote by  $\langle v, u \rangle$ , the standard inner product between the vectors  $u, v$ . For matrices  $U, V \in \mathbb{R}^{d \times d}$ , we use  $\langle U, V \rangle$  to denote the trace inner product  $\sum_{ij} U_{ij} V_{ij}$ . For a matrix  $A \in \mathbb{R}^{d \times d}$ , we use  $A^\flat$  to denote the flattened vector in  $\mathbb{R}^{d^2}$ , and for a  $v \in \mathbb{R}^{d^2}$ , we use  $v^\sharp$  to denote the unique matrix  $A$  such that  $A^\flat = v^\sharp$ . For a matrix  $A$ , we let  $A^\dagger$  denote its pseudo-inverse. We use  $\otimes$  to denote the Kronecker product. For a matrix  $A$ , we use  $\ker(A)$  for the null space of  $A$ . We use  $X \sim D$  to denote that a random variable  $X$  is distributed according to the distribution  $D$ . We use  $\mathcal{N}(\mu, \Sigma)$  for the Gaussian distribution with mean  $\mu$  and covariance matrix  $\Sigma$ . For a set  $S$ , we use  $X \sim S$  to denote that  $X$  is distributed uniformly at random from  $S$ . We use  $a \lesssim b$  to denote that there exists an absolute universal constant  $C > 0$  (independent of the variables or parameters on which  $a$  and  $b$  depend) such that  $a \leq C b$ .
2.1 Deterministic Conditions
Our algorithm will rely on the uncorrupted inliers satisfying a set of properties, similar to the "stability conditions" from [DKKLMS16]. Intuitively, these are are concentration properties for sets of samples, but with the added requirement that every large subset of the samples also satisfies these properties.
Definition 2.1  $((\eta, \epsilon)$ -Stable Set). Let  $D$  be a distribution with mean  $\mu$  and covariance  $\Sigma$ . We say a set of points  $A \subset \mathbb{R}^d$  is  $(\eta, \epsilon)$ -stable with respect to  $D$ , if for any subset  $A' \subseteq A$  with  $|A'| \geq (1 - \epsilon)|A|$ , the following hold: for every  $v \in \mathbb{R}^d$ , symmetric  $U \in \mathbb{R}^{d \times d}$ , and every even degree-2 polynomial  $p$ :
(L.1)  $\left|\frac{1}{|A'|}\sum_{x\in A'}v^\top (x - \mu)\right|\leq 0.1\sqrt{(v^\top\Sigma v)}.$
(L.2)  $\left|\left\langle \frac{1}{|A'|} \sum_{x \in A'} (x - \mu)(x - \mu)^{\top} - \Sigma, U \right\rangle \right| \leq 0.1 \left\| \Sigma^{1/2} U \Sigma^{1/2} \right\|_{\mathrm{F}}$ .
(L.3)  $\mathbf{Pr}_{X\sim A^{\prime}}\left[\big|p(X) - \mathbf{E}_{X\sim D}[p(X)]\big| > 10\sqrt{\frac{\mathbf{Var}}{Y\sim D} [p(Y)]}\ln \left(\frac{2}{\eta}\right)\right]\leq \eta .$
(L.4)  $\mathbf{Var}_{X\sim A'}[p(X)]\leq 4\mathbf{Var}_{X\sim D}[p(X)].$
(L.5) The null space of second moment matrix of  $A'$  is contained in the null space of  $\Sigma$ , i.e.,  $\ker \left( \sum_{x \in A'} xx^\top \right) \subseteq \ker(\Sigma)$ .
A Gaussian dataset is "stable" with high probability [DKKLMS16]; formally, we have Lemma 2.2, proved in Appendix D.1. Moreover, Lemma 2.2 can be extended to a variety of distributions (cf. Remark D.1).
Lemma 2.2 (Deterministic Conditions Hold with High Probability). For a sufficiently small positive constant  $\epsilon_0$  and a sufficiently large absolute constant  $C$ , a set of  $m > Cd^2\log^5(d / (\eta \delta)) / \eta^2$  samples from  $\mathcal{N}(\mu, \Sigma)$ , with probability  $1 - \delta$ , is  $(\eta, \epsilon)$ -stable set with respect to  $\mu, \Sigma$  for all  $\epsilon \leq \epsilon_0$ .
3 Analysis of a Single Recursive Call of the Algorithm
Our algorithm, Algorithm 1, filters and splits samples into multiple sets recursively, until we can certify that the empirical second moment matrix of the "current data set" is suitable to be included in the returned list of covariances. As a reference point, we define the notations and assumptions necessary to analyze each recursive call below. However, before moving to the formal analysis we will first give an informal overview of the algorithm's steps and the high-level ideas behind them.
Assumption 3.1 (Assumptions and notations for a single recursive call of the algorithm).
-  $S = \{x_{i}\}_{i=1}^{n}$  is a set of  $n$  uncontaminated samples, which is assumed to be  $(\eta, 2\epsilon_0)$ -stable with respect to the inlier distribution  $D$  having mean and covariance  $\mu, \Sigma$  (c.f. Definition 2.1). We assume  $\eta \leq 0.001$ ,  $\epsilon_0 = 0.01$ , and  $\mathbf{Var}_{X \sim D}[X^{\top}AX] \leq C_1(\| \Sigma^{1/2}A\Sigma^{1/2}\|_F^2 + \| \Sigma^{1/2}A\mu\|_2^2)$  for all symmetric  $d \times d$  matrices  $A$  and a constant  $C_1$ .
-  $T$  is the input set to the current recursive call of the algorithm (after the adversarial corruptions), which satisfies  $|S \cap T| \geq (1 - 2\epsilon_0)|S|$  and  $|T| \leq (1 / \alpha)|S|$ .
We denote  $H = \mathbf{E}_{X\sim T}\left[XX^{\top}\right]$
- We denote by  $\widetilde{S}, \widetilde{T}$  the versions of  $S$  and  $T$  normalized by  $H^{\dagger/2} \colon \widetilde{S} = \{H^{\dagger/2}x : x \in S\}$  and  $\widetilde{T} = \{H^{\dagger/2}x : x \in T\}$ . We use the notation  $\widetilde{x}$  for elements in  $\widetilde{S}$  and  $\widetilde{T}$ , and  $x$  for elements in  $S$  and  $T$ . Similarly, we use the notation  $\widetilde{X}$  for random variables with support in  $\widetilde{S}$  or  $\widetilde{T}$ .
- The mean and covariance of the inlier distribution  $D$  after transformation with  $H^{\dagger /2}$  are denoted by  $\tilde{\mu} \coloneqq H^{\dagger /2}\mu$ ,  $\widetilde{\Sigma} \coloneqq H^{\dagger /2}\Sigma H^{\dagger /2}$ . We denote the empirical mean and covariance of the transformed inliers in  $T$  by  $\hat{\mu} \coloneqq \mathbf{E}_{\widetilde{X} \sim \widetilde{S} \cap \widetilde{T}}[\widetilde{X}]$  and  $\hat{\Sigma} \coloneqq \mathbf{Cov}_{\widetilde{X} \sim \widetilde{S} \cap \widetilde{T}}[\widetilde{X}]$ .
Much of the algorithm uses the fact that, for a Gaussian, even quadratic polynomials have a small variance. We will leverage this for filtering and clustering samples. See Appendix E for the proof.
Lemma 3.2. Make Assumption 3.1 and recall that  $H = \mathbf{E}_{X \sim T}[XX^{\top}]$ , where  $T$  is the corrupted version of a stable inlier set  $S$ . For every symmetric matrix  $A$  with  $\| A \|_{\mathrm{F}} = 1$ , we have that  $\operatorname{Var}_{X \sim D}[(H^{\dagger/2}X)^{\top}A(H^{\dagger/2}X)] \leq 18C_1 / \alpha^2$ .
Armed with Lemma 3.2, we can now give a high-level overview of a recursive call of Algorithm 1:
1. In our notation, we call the current data set  $T$ . Denoting  $H = \mathbf{E}_{X \sim T}[XX^{\top}]$  for its the empirical second moment matrix, we construct the normalized data set  $\widetilde{T} = \{H^{-1/2}x : x \in T\}$ . The normalization allows us to bound the covariance  $\Sigma$  in terms of  $H$ .
2. Since we are trying to estimate a covariance, consider the vectors  $\tilde{s} = \{(\tilde{x}\tilde{x}^{\top})^{\flat}:\tilde{x}\in \widetilde{T}\}$ , which are the second moment matrices of each data point flattened into vectors.
3. The first step is standard in filtering-based outlier-robust estimation: we test whether the covariance of the  $\tilde{s}$  vectors is small. If so, we are able to prove that the current  $H$  is a good approximate square root of  $\Sigma$  (c.f. Section 3.1) hence we just return  $H$ .
4. If the first test fails, that would imply that the empirical covariance of the  $\tilde{s}$  vectors is large in some direction. We want to leverage this direction to make progress, either by removing outliers through filtering or by bi-partitioning our samples into two clear clusters.
5. To decide between the 2 options, consider projecting the  $\tilde{s}$  vectors onto their largest variance direction. Specifically, let  $A$  be the matrix lifted from the largest eigenvector of the covariance of the  $\tilde{s}$  vectors. Define the vectors  $\tilde{y} = \tilde{x}^{\top}A\tilde{x} = \langle \tilde{x}\tilde{x}^{\top},A\rangle$  for  $\tilde{x}\in \widetilde{T}$ , corresponding to the 1-dimensional projection of  $\tilde{s}$  onto the  $A^{\flat}$  direction. Since we have failed the first test, these  $\tilde{y}$  elements must have a large variance. We will decide to filter or divide our samples, based on whether the  $\alpha m$ -smallest and  $\alpha m$ -largest elements of the  $\tilde{y}$ s are close to each other.
6. If they are close, yet we have large variance, we will use this information to design a score function and perform filtering that removes a random sample with probability proportional to its score. We will then go back to Step 1. This would work because by Lemma 3.2 and by stability (Definition 2.1), the (unfiltered) inliers have a small empirical variance within themselves, meaning that the large total empirical variance is mostly due to the outlier.
Ideally, the score of a sample would be (proportional to) the squared distance between the sample and the mean of the inliers—the total inlier score would then be equal to the inlier variance. However, since we do not know which points are the inliers, we instead use the median of all the projected samples as a proxy for the unknown inlier mean. We show that the distance between the  $\alpha m$ -smallest and largest  $\tilde{y}$ s bounds the difference between the ideal and proxy scores.
7. Otherwise,  $\alpha m$ -smallest and  $\alpha m$ -largest elements of the  $\tilde{y}$ s are far apart. By the stability condition Definition 2.1 (specifically, Condition (L.3)), most of the inliers must be close to each other under this 1-dimensional projection. Therefore, the large quantile range necessarily means there is a threshold under this projection to divide the samples into two sets, such that each set has at least  $\alpha m$  points and most of the inliers are kept within a single set.
The score function mentioned in Step 6 upper bounds the maximum variance we check in Step 3. For simplicity, in the actual algorithm (Algorithm 1) we use the score directly for the termination check instead of checking the covariance, but it does not matter technically which quantity we use.
Remark 3.3 (Runtime of Algorithm 1). We claim that each "loop" in Algorithm 1 takes  $\tilde{O}(md^2)$  time to compute. The number of times we run the "loop" is at most  $O(m)$ , since each loop either ends in termination, removes 1 element from the dataset, or splits the dataset, all of which can happen at most  $O(m)$  times. From this, we can conclude a runtime of  $\tilde{O}(m^2 d^2)$ . The sample complexity of our algorithm is also explicitly calculable to be  $\tilde{O}(d^2 / \alpha^6)$ , which follows from Lemma 2.2 and the choice of parameter  $\eta = \Theta(\alpha^3)$  from Theorem F.1 (the formal version of Theorem 1.2).
To see the runtime of a single loop: the most expensive operations in each loop are to compute  $H_{t}$ , its pseudo-inverse, and to compute the symmetric matrix  $A$  in Line 7 that is the top eigenvector of a  $d^{2} \times d^{2}$  matrix. Computing  $H_{t}$  trivially takes  $O(md^{2})$  time, resulting in a  $d \times d$  matrix. Its pseudoinverse can be computed in  $O(d^{\omega})$  time, which is dominated by  $O(md^{2})$  since  $m \gtrsim d$ . Lastly, we observe that, instead of actually computing the top eigenvector in Line 7 to yield the matrix  $A$ , it suffices in our analysis to compute a matrix  $B$  whose Rayleigh quotient  $(B^{\flat})^{\top}\left(\mathbf{Cov}_{\widetilde{X} \sim \widetilde{T}_t}[\widetilde{X}^{\otimes 2}]\right)B^{\flat} / ((B^{\flat})^{\top}B^{\flat})$  is at least  $\frac{1}{2}$  times  $(A^{\flat})^{\top}\left(\mathbf{Cov}_{\widetilde{X} \sim \widetilde{T}_t}[\widetilde{X}^{\otimes 2}]\right)A^{\flat} / ((A^{\flat})^{\top}A^{\flat})$ . We can do this via  $O(\log d)$  many
Algorithm 1 List-Decodable Covariance Estimation in Relative Frobenius Norm
1: Constants:  $m, \alpha, \eta, R \coloneqq C(1 / \alpha^2) \log (1 / (\epsilon_0 \alpha))$  for  $C > 6000\sqrt{C_1}$ ,  $C' > 720 / \epsilon_0$  (where  $C_1, \epsilon_0$  are defined in Assumption 3.1).  
2: function COVLISTDECODING( $T_0$ )  
3:  $t \gets 0$ .  
4: loop  
5: Compute  $H_t = \mathbf{E}_{X \sim T_t}[XX^\top]$ .  
6: Let  $\widetilde{T}_t = \{H_t^{\dagger/2}x : x \in T_t\}$  be the transformed set of samples.  
7: Let  $A$  be the symmetric matrix corresponding to the top eigenvector of  $\mathrm{Cov}_{\widetilde{X} \sim \widetilde{T}_t}[\widetilde{X}^{\otimes 2}]$ .  
8:Normalize  $A$  so that  $\|A\|_{\mathrm{F}} = 1$ .  
9: Compute the set  $\widetilde{Y}_t = \{\widetilde{x}^\top A\widetilde{x} : \widetilde{x} \in \widetilde{T}_t\}$   
10:Compute the  $\alpha m/9$ -th smallest element  $q_{\text{left}}$  as well as the  $\alpha m/9$ -th largest element  $q_{\text{right}}$ , as well as the median  $\widetilde{y}_{\text{median of }}(\widetilde{Y}_t)$ .  
11:Define the function  $f(\widetilde{x}) = (\widetilde{x}^\top A\widetilde{x} - \widetilde{y}_{\text{median}})^2$ .  
12:if  $\mathbf{E}_{\widetilde{X} \sim \widetilde{T}_t}[f(\widetilde{X})] \leq C'R^2/\alpha^3$  then ▷c.f. Lemma 3.4  
13:If  $|T_t| \geq 0.5\alpha m$  then return  $\{T_t\}$  else return the empty list.  
14:else if  $q_{\text{right}} - q_{\text{left}} \leq R$  then ▷c.f. Lemma 3.5  
15:Let the probability mass function  $p(\widetilde{x}) := f(\widetilde{x}) / \sum_{\widetilde{x} \in \widetilde{T}_t} f(\widetilde{x})$ .  
16:Pick  $x_{\text{removed}} \in T_t$  according to  $p(H_t^{\dagger/2}x)$ .  
17:  $T_{t+1} \gets T_t \setminus \{x_{\text{removed}}\}$ .  
18:else  
19:  $\tau \gets \text{FindDivider}(\widetilde{Y}_t, \alpha m/9)$ . ▷c.f. Lemma 3.6  
20:  $T' \gets \{H_t^{1/2}\widetilde{x} : \widetilde{x} \in \widetilde{T}_t, \widetilde{x}^\top A\widetilde{x} \leq \tau\}, T'' \gets \{H_t^{1/2}\widetilde{x} : \widetilde{x} \in T_t, \widetilde{x}^\top A\widetilde{x} > \tau\}$ .  
21:L1← COVLISTDECODING( $T'$ ), L2← COVLISTDECODING( $T''$ ).  
22:return L1 ∪ L2.  
23:t ← t + 1.
power iterations. Since
$$
\underset {\tilde {X} \sim \tilde {T} _ {t}} {\mathbf {C o v}} [ \widetilde {X} ^ {\otimes 2} ] = \frac {1}{| \widetilde {T} _ {t} |} \sum_ {z \in \tilde {T} _ {t}} z z ^ {\top} - \left(\frac {1}{| \widetilde {T} _ {t} |} \sum_ {z \in \tilde {T} _ {t}} z\right) ^ {\top} \left(\frac {1}{| \widetilde {T} _ {t} |} \sum_ {z \in \tilde {T} _ {t}} z\right),
$$
we can compute each matrix-vector product in  $O(|T_t|d^2) \leq O(md^2)$  time, thus yielding an  $\tilde{O}(m^2 d^2)$  runtime for the power iteration.
3.1 Certificate Lemma: Bounded Fourth Moment Implies Closeness
The first component of the analysis is our certificate lemma, which states that, if the empirical covariance of the (flattened) second moment matrices of current data set (after normalization) is bounded, then the empirical second moment matrix  $H$  of the current data set is a good approximation to the covariance of the Gaussian component we want to estimate.
Lemma 3.4 (Case when we stop and return). Make Assumption 3.1. Let  $w \in \mathbb{R}^{d^2}$  be the leading eigenvector of the  $\mathbf{Cov}_{\widetilde{X} \sim \widetilde{T}}[\widetilde{X}^{\otimes 2}]$  with  $\| w \|_2 = 1$ , and let  $A \in \mathbb{R}^{d \times d}$  be  $w^\sharp$ . Note that  $\| w \|_2 = 1$  implies  $\| A \|_F = 1$ . Then, we have  $\left\| H^{\dagger/2} \Sigma H^{\dagger/2} - I \right\|_F^2 \lesssim (1/\alpha) \mathbf{Var}_{\widetilde{X} \sim \widetilde{T}}[\widetilde{X}^\top A \widetilde{X}] + 1/\alpha^2$ .
See Appendix E.2 for the proof. Our termination check of Line 12 uses the score  $f(\widetilde{x}) = (\widetilde{x}^{\top}A\widetilde{x} - \widetilde{y}_{\mathrm{median}})^{2}$  where  $\widetilde{y}_{\mathrm{median}}$  is the median of  $\{\widetilde{x}^{\top}A\widetilde{x} : \widetilde{x} \in \widetilde{T}\}$ . Since  $\mathbf{Var}_{\widetilde{X} \sim \widetilde{T}}[\widetilde{X}^{\top}A\widetilde{X}] \leq \mathbf{E}_{\widetilde{X} \sim \widetilde{T}}[f(\widetilde{X})]$  our check ensures that  $\mathbf{Var}_{\widetilde{X} \sim \widetilde{T}}[\widetilde{X}^{\top}A\widetilde{X}] \leq \mathrm{poly}(1 / \alpha)$  before returning.
3.2 Filtering: Removing Extreme Outliers
As discussed in the algorithm outline, if the termination check fails, namely the expected score over the entire set of  $\widetilde{T}$  is large, then we proceed to either filter or bi-partition our samples. This subsection
states the guarantees of the filtering procedure, which assumes that the  $\alpha m$ -smallest and largest elements in the set  $\{\tilde{x}^\top A\tilde{x} : \tilde{x} \in \tilde{T}\}$  have distance at most  $R$  for some  $R = \mathrm{poly}(1 / \alpha)$ .
Recall from Lemma 3.2 that  $\mathbf{Var}_{\widetilde{X}\sim H^{\dagger /2}D}[\widetilde{X}^{\top}A\widetilde{X}]$  =  $\mathbf{E}_{\widetilde{X}\sim H^{\dagger /2}D}[(\widetilde{X}^{\top}A\widetilde{X}-$ $\mathbf{E}_{\widetilde{X}\sim H^{\dagger /2}D}[\widetilde{X}^{\top}A\widetilde{X}])^{2}]$  is bounded by  $O(1 / \alpha^{2})$ . By stability, the same is true for  $\mathbf{E}_{\widetilde{X}\sim \widetilde{S}\cap \widetilde{T}}[(\widetilde{X}^{\top}A\widetilde{X} -\mathbf{E}_{\widetilde{X}\sim \widetilde{S}\cap \widetilde{T}}[\widetilde{X}^{\top}A\widetilde{X} ])^{2}]$ . Notice that this looks almost like our score function  $f(\widetilde{X})$ , except that in  $f(\widetilde{X})$  we use  $\widetilde{y}_{\mathrm{median}}$  for centering instead of  $\mathbf{E}_{\widetilde{X}\sim \widetilde{S}\cap \widetilde{T}}[\widetilde{X}^{\top}A\widetilde{X} ]$ , since the latter quantity is by definition unknown to the algorithm. In Lemma E.3, we show that the two quantities have distance upper bounded by  $O(R)$ , where  $R$  is the quantile distance in our assumption, which in turn implies that the inliers in  $\widetilde{S}\cap \widetilde{T}$  contribute very little to  $\mathbf{E}_{\widetilde{X}\sim \widetilde{T}}[f(\widetilde{X})]$ . Given that  $\mathbf{E}_{\widetilde{X}\sim \widetilde{T}}[f(\widetilde{X})]$  is large, by virtue of having failed the termination check, we can then conclude that most of the score contribution comes from the outliers. Thus, we can safely use the score to randomly pick an element in the dataset for removal, with probability proportional to its score, and the element will be overwhelmingly more likely to be an outlier rather than an inlier. Lemma 3.5 below states the precise guarantees on the ratio of the total score of the inliers versus the total score over the entire dataset.
Lemma 3.5 (Filtering). Make Assumption 3.1. Let  $A$  be an arbitrary symmetric matrix with  $\| A \|_{\mathrm{F}} = 1$ . Let  $R = C(1 / \alpha) \log(1 / \eta)$  for  $C \geq 100 \sqrt{C_1}$ . Define  $\widetilde{y}_{\mathrm{median}} = \mathrm{Median}(\{\widetilde{x}^{\top} A \widetilde{x} \mid \widetilde{x} \in \widetilde{T}\})$ . Define the function  $f(\widetilde{x}) := (\widetilde{x}^{\top} A \widetilde{x} - \widetilde{y}_{\mathrm{median}})^2$ . Let  $m_1$  be a number less than  $|S|/3$ . Denote by  $q_i$  the  $i$ -th smallest point of  $\{\widetilde{x}^{\top} A \widetilde{x} \mid \widetilde{x} \in \widetilde{T}\}$ . If  $q_{|T|-m_1} - q_{m_1} \leq R$  and  $\mathbf{E}_{X \sim T}[f(x)] > C'R^2/\alpha^3$  for  $C' \geq 720/\epsilon_0$ , that is, in the case where the check in Line 12 fails, then, the function  $f(\cdot)$  satisfies  $\sum_{\widetilde{x} \in \widetilde{T}} f(\widetilde{x}) > \frac{40}{\epsilon_0} \frac{1}{\alpha^3} \sum_{\widetilde{x} \in \widetilde{S} \cap \widetilde{T}} f(\widetilde{x})$ .
The score ratio determines (in expectation) the ratio between the number of outliers and inliers removed. In Lemma 3.5, the ratio is in the order of  $1 / \alpha^3$  —this will allow us to guarantee that at the end of the entire recursive execution of Algorithm 1, we would have removed at most a  $0.01\alpha$  fraction of the inliers. See Remark F.5 in Appendix F for more details.
3.3 Divider: Identifying Multiple Clusters and Recursing
The previous subsections covered the cases where (i) the expected score is small, or (ii) the expected score over  $\widetilde{T}$  is large and the  $\alpha$  and  $1 - \alpha$  quantiles of  $\{\widetilde{X}^{\top}A\widetilde{X}:\widetilde{x}\in \widetilde{T}\}$  are close to each other. What remains is the case when both the expected score is large yet the quantiles are far apart. In this instance, we will not be able to make progress via filtering using the above argument. This is actually an intuitively reasonable scenario, since the outliers may in fact have another  $\approx \alpha m$  samples that are distributed as a different Gaussian with a very different covariance—the algorithm would not be able to tell which Gaussian is supposed to be the inliers. We will argue that, when both the expected score is large and the quantiles are far apart, the samples are in fact easy to bipartition into 2 clusters, such that the most of the inliers fall within 1 side of the bipartition. This allows us to make progress outside of filtering, and this clustering mechanism also allows us to handle Gaussian mixture models and make sure we (roughly) handle components separately.
The key intuition is that, by the stability Conditions (L.3) and (L.4), we know that the inliers under the 1-dimensional projection  $\{\widetilde{X}^{\top}A\widetilde{X}:\widetilde{X}\in \widetilde{S}\cap \widetilde{T}\}$  must be well-concentrated, in fact lying in an interval of length  $\widetilde{O}(1 / \alpha)$ . The fact that the quantile range is wide implies that there must be some point within the range that is close to very few samples  $\widetilde{X}^{\top}A\widetilde{X}$ , by an averaging argument. We can then use the point as a threshold to bipartition our samples. See below for the precise statement.
Algorithm 2 Divider for list decoding
1: function FINDDIVIDER  $(T,n^{\prime},m_{1})$
2: Let the  $m_{1}$ -th smallest point be  $q_{m_1}$  and  $m_{1}$ -th largest point be  $q_{|T| - m_1}$ .
3: Divide the interval  $[q_{m_1}, q_{|T|-m_1}]$  into  $2m'/n'$  equally-sized subintervals.
4: Find a subinterval  $I'$  with at most  $n'/2$  points and return its midpoint.
Lemma 3.6 (Divider Algorithm). Let  $T = (y_{1}, \ldots, y_{m'})$  be a set of  $m'$  points in  $\mathbb{R}$ . Let  $S_{\mathrm{proj}} \subset T$  be a set of  $n'$  points such that  $S_{\mathrm{proj}}$  is supported on an interval  $I$  of length  $r$ . For every  $i \in [m']$ , let
the  $i$ -th smallest point of the set  $T$  be  $q_{i}$ . Suppose  $q_{|T|-m_{1}} - q_{m_{1}} \geq R$  such that  $R \geq 10(m' / n')r$ . Then, given  $T$  and  $n'$ ,  $m_{1}$ , Algorithm 2 returns a point  $t$  such that if we define  $T_{1} = \{x \in T : x \leq t\}$  and  $T_{2} = T \setminus T_{1}$  then: (i)  $\min(|T_{1}|, |T_{2}|) \geq m_{1}$  and (ii)  $S_{\mathrm{proj}} \subseteq T_{1}$  or  $S_{\mathrm{proj}} \subseteq T_{2}$ .