SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Here  $\sigma$  denotes the randomness used by the algorithm  $\mathcal{A}'$ . Now, if  $\mathbf{X}$  is a random matrix constructed as  $(\alpha/\sqrt{n}) \sum_{i=1}^{r} \mathbf{u}_i \mathbf{v}_i^{\top}$  with  $\mathbf{u}_i, \mathbf{v}_i$  being random vectors with independent Gaussian entries of mean 0 and variance 1, then with probability  $\geq 99/100$ ,  $\| \mathbf{X} \|_{\mathsf{F}}^2 = \Theta(nr)$ . Thus,  $\operatorname{Pr}_{\mathbf{X}, \mathbf{G}, \sigma}[\mathcal{A}'(\mathbf{X} + \mathbf{G}, \sigma)$  is correct]  $\geq 8/10$ . Hence, there exists some fixed  $\sigma$  such that
$$
\operatorname * {P r} _ {\boldsymbol {X}, \boldsymbol {G}} \left[ \mathcal {A} ^ {\prime} (\boldsymbol {X} + \boldsymbol {G}, \sigma) \text {i s c o r r e c t} \right] \geq 8 / 1 0.
$$
Thus, the existence of a randomized algorithm that solves low rank matrix recovery as in Theorem 1.1 implies the existence of a deterministic algorithm which given access to perfect linear measurements of random matrix  $(\alpha/\sqrt{n}) \sum_{i=1}^{r} \boldsymbol{u}_i \boldsymbol{v}_i^{\mathsf{T}} + G$  outputs a reconstruction of  $(\alpha/\sqrt{n}) \sum_{i=1}^{r} \boldsymbol{u}_i \boldsymbol{v}_i^{\mathsf{T}}$  with probability  $\geq 8/10$ . This is essentially a reduction from randomized algorithms to deterministic algorithms using Yao's lemma. From here on, we prove lower bounds on such deterministic algorithms and conclude the lower bounds in Theorem 1.1. For simplicity, we explain the proof of our lower bound for  $r = 1$  here and extend to general  $r$  later. We consider the random matrix  $\boldsymbol{G} + (\alpha/\sqrt{n}) \boldsymbol{u} \boldsymbol{v}^{\mathsf{T}}$  and show how the lower bound proof proceeds.
Note that the matrix  $\mathbf{A} \coloneqq \mathbf{G} + (\alpha / \sqrt{n})\mathbf{u}\mathbf{v}^{\top} \in \mathbb{R}^{n \times n}$  can be flattened to the vector  $\mathbf{a} = \mathbf{g} + (\alpha / \sqrt{n})(\mathbf{u} \otimes \mathbf{v}) \in \mathbb{R}^{n^2}$ . Also, a general linear measurement, which we call a query  $Q \in \mathbb{R}^{n \times n}$ , can be vectorized to  $q = \operatorname{vec}(Q) \in \mathbb{R}^{n^2}$  with  $Q(\mathbf{A}) = \langle q, \mathbf{a} \rangle$ . Fix any deterministic algorithm. In the first round, the algorithm starts with a fixed matrix  $Q^{(1)} \in \mathbb{R}^{k \times n^2}$  that corresponds to the  $k$  queries and receives the response  $r^{(1)} := Q^{(1)}\mathbf{a}$ . Then, as a function of the response  $r^{(1)}$  in the first round, the algorithm picks a matrix  $Q^{(2)}[r^{(1)}]$  in the second round and receives the response  $r^{(2)} := Q^{(2)}[r^{(1)}] \cdot \mathbf{a}$ . Similarly, in the  $i$ -th round, the deterministic algorithm picks a matrix  $Q^{(i)}[r^{(1)}, \ldots, r^{(i-1)}] \in \mathbb{R}^{k \times n^2}$  as a function of  $r^{(1)}, \ldots, r^{(i-1)}$  and receives the response  $r^{(i)} := Q^{(i)}[r^{(1)}, \ldots, r^{(i-1)}] \cdot \mathbf{a}$ . Note that we assumed that the query matrices  $Q^{(i)}$  chosen by the algorithm have orthonormal rows and also that  $Q^{(i)}(Q^{(j)})^{\top} = 0$ , i.e., the queries across rounds are also orthonormal.
For a fixed  $u, v \in \mathbb{R}^n$ , we see that the response  $\pmb{r}^{(1)} = Q^{(1)}\pmb{g} + (\alpha/\sqrt{n})Q^{(1)}(u \otimes v)$ . As the matrix  $Q^{(1)}$  has orthonormal rows, the random variable  $Q^{(1)}\pmb{g} \equiv \pmb{g}_k$ , where  $\pmb{g}_k \sim N(0, I_k)$  is drawn from a mean-zero normal distribution with identity covariance. Thus, for fixed  $u, v$ , the distribution of the first round responses to the algorithm is  $N((\alpha/\sqrt{n})Q^{(1)}(u \otimes v), I_k)$ . Now the key observation is that  $\| (\alpha/\sqrt{n})Q^{(1)}(\pmb{u} \otimes \pmb{v}) \|_2^2 = \Theta (\alpha^2 k/n)$  with high probability over the inputs  $(\pmb{u}, \pmb{v})$ . This uses a recent concentration result for random tensors due to [31], and critically uses the fact that  $Q^{(1)}$  has operator norm 1 and Frobenius norm  $\sqrt{k}$ . This means that for a large fraction of  $(u, v)$  pairs, the distribution of the responses seen by the algorithm in the first round is close to  $N(0, I_k)$ , and therefore just looking at the response  $\pmb{r}^{(1)}$ , the algorithm cannot have a lot of "information" about which  $(u, v)$  pair is involved in the matrix that is unknown to the algorithm. So the query chosen in the second round  $Q^{(2)}[\pmb{r}^{(1)}]$  cannot have a "large" value of  $Q^{(2)}[\pmb{r}^{(1)}](u \otimes v)$  for most inputs  $(u, v)$ , with a high probability over the Gaussian component of the matrix.
Suppose the value of  $Q^{(2)}[r^{(1)}](u \otimes v)$  is small. Again,  $r^{(2)} = Q^{(2)}[r^{(1)}]g + (\alpha/\sqrt{n})Q^{(2)}[r^{(1)}](u \otimes v)$ . Crucially, as the queries in round 2 are orthogonal to the queries in round 1, we have by the rotational invariance of the Gaussian distribution that  $Q^{(2)}[r^{(1)}]g$  is independent of  $Q^{(1)}g$ , and that  $Q^{(2)}[r^{(1)}]g$  is distributed as  $N(0, I_k)$ . So, for a fixed  $u, v$ , conditioned on the first round response  $r^{(1)}$ , the distribution of the second round response  $r^{(2)}$  is given by  $N((\alpha/\sqrt{n})Q^{(2)}[r^{(1)}](u \otimes v), I_k) \approx N(0, I_k)$  using the assumption that  $Q^{(2)}[r^{(1)}](u \otimes v)$  is small. We again have that for a large fraction of pairs  $(u, v)$  for which  $Q^{(2)}[r^{(1)}](u \otimes v)$  is small, the distribution of the second round response is also close to  $N(0, I_k)$ . Therefore, the algorithm again does not gain a lot of information about which  $(u, v)$  pair is involved in the matrix, and the third round query  $Q^{(3)}[r^{(1)}, r^{(2)}]$  cannot have a large value of  $\|Q^{(3)}[r^{(1)}, r^{(2)}](u \otimes v)\|_2$ . The proof proceeds similarly for further rounds.
To formalize the above intuitive idea, we use Bayes risk lower bounds [7]. We show that with a large probability over the input matrix, the squared projection of  $\pmb{u} \otimes \pmb{v}$  onto the query space of the algorithm is upper bounded and we use an iterative argument to show that an upper bound on the information up until round  $i$  can in turn be used to upper bound the information up until round  $i + 1$ . Bayes risk bounds are very general and let us obtain upper bounds on the information learned by a deterministic learner. Concretely, let  $\Theta$  be a parameter space and  $\mathcal{P} = \{P_{\theta} : \theta \in \Theta\}$  be a set
of distributions, one for each  $\theta \in \Theta$ . Let  $w$  be a distribution over  $\Theta$ . We sample  $\theta \sim w$  and then  $x \sim P_{\theta}$  and provide the learner with  $x$ . Given an action space  $\mathcal{A}$ , the learner uses a deterministic decision rule  $\mathfrak{d} : \mathcal{X} \to \mathcal{A}$  to minimize a 0-1 loss function  $L : \Theta \times \mathcal{A} \to \{0,1\}$  in expectation, i.e.,  $\operatorname{E}_{\theta \sim w}[\operatorname{E}_{x \sim P_{\theta}}L(\theta, \mathfrak{d}(x))]$ . Let  $R_{\mathrm{Bayes}}(L, \Theta, w) = \inf_{\mathfrak{d}} E_{\theta \sim w}[E_{x \sim P_{\theta}}L(\theta, \mathfrak{d}(x))]$  be the loss achievable by the best deterministic decision rule  $\mathfrak{d}$ . Bayes risk lower bounds let us obtain lower bounds on  $R_{\mathrm{Bayes}}$ .
In our setting after round 1, we have  $\Theta = \{(u,v):u,v\in \mathbb{R}^n\}$ ,  $w$  is the joint distribution of two independent standard Gaussian random variables in  $\mathbb{R}^n$  and for each  $(u,v)\in \Theta$  we let  $P_{uv}$  be the distribution of  $\pmb{r}^{(1)} = \pmb{g}_k + (\alpha /\sqrt{n})Q^{(1)}(u\otimes v)$ , an action space  $\mathcal{A}$  of all  $k\times n^2$  matrices with orthonormal rows (corresponding to the queries in the next round), and define a 0-1 loss function
$$
L ((u, v), Q) = \left\{ \begin{array}{l l} 0 & \text {i f} \| Q (u \otimes v) \| _ {2} ^ {2} \geq T \\ 1 & \text {i f} \| Q (u \otimes v) \| _ {2} ^ {2} <   T \end{array} \right.
$$
for an appropriate threshold parameter  $T$ . By setting  $T$  appropriately as a function of  $t$ , we obtain a Bayes risk lower bound of  $R_{\mathrm{Bayes}} \geq 1 - 1 / (100t^2)$ . Thus, we obtain that for any deterministic decision rule  $\mathfrak{d}$ , with probability  $\geq 1 - 1 / 10t$  over  $(\mathbf{u}, \mathbf{v}) \sim w$ , we have
$$
\operatorname {E} _ {\boldsymbol {r} ^ {(1)} \sim P _ {\boldsymbol {u v}}} [ L ((\boldsymbol {u}, \boldsymbol {v}), \mathfrak {d} (\boldsymbol {r} ^ {(1)})) ] \geq 1 - 1 / 1 0 t
$$
and in particular, we have that with probability  $\geq 1 - 1 / 10t$  over  $(\pmb{u},\pmb{v})\sim w$
$$
\Pr_ {\boldsymbol {r} ^ {(1)} \sim P _ {\boldsymbol {u v}}} [ \| Q ^ {(2)} [ \boldsymbol {r} ^ {(1)} ] (\boldsymbol {u} \otimes \boldsymbol {v}) \| _ {2} ^ {2} <   T ] \geq 1 - 1 / 1 0 t. \tag {2}
$$
The above statement essentially says that with probability  $\geq 1 - 1/10t$  over the inputs  $(\boldsymbol{u}, \boldsymbol{v})$ , the second query  $Q^{(2)}[\boldsymbol{r}^{(1)}]$  chosen by the deterministic algorithm has the property that  $\|Q^{(2)}[\boldsymbol{r}^{(1)}](\boldsymbol{u} \otimes \boldsymbol{v})\|_2^2 < T$  with probability  $\geq 1 - 1/10t$  over the Gaussian  $\pmb{G}$ . In the second round, we restrict our analysis of the algorithm to only those  $(u, v)$  which satisfy (2). We again define a distribution  $w'$  over the inputs and for each such  $(u, v)$  we define a distribution  $P_{uv}$  over the round 1 and round 2 responses received by the algorithm. We define a new loss function with parameter  $T' = \Delta \cdot T$  for a multiplicative factor  $\Delta$  and again obtain a statement similar to (2) for a large fraction of inputs  $(u, v)$  and repeat a similar argument for  $t$  rounds and show that there is an  $\Omega(1)$  fraction of the inputs for which the squared norm of the projection of  $u \otimes v$  onto the query space after  $t$  rounds is bounded by  $T(t)$  with high probability over the Gaussian part of the input. This gives the result in Theorem 3.2. Note that  $\| \boldsymbol{u} \otimes \boldsymbol{v} \|_2^2 = \Omega(n^2)$  with high probability and for any fixed matrix  $Q$  with  $k$  orthonormal rows,  $\operatorname{E}[\| Q(\boldsymbol{u} \otimes \boldsymbol{v}) \|_2^2] = k$  which corresponds to the amount of "information" the algorithm starts with. As we show that the "information" in each round grows by some multiplicative factor  $\Delta$ , a number  $\Omega(\log_{\Delta}(n^2 / k))$  of rounds is required to obtain an "information" of  $\Theta(n^2)$ , which is how we obtain our lower bounds. Here information is measured as the squared projection of  $\boldsymbol{u} \otimes \boldsymbol{v}$  onto the query space of the algorithm.
We also extend our results to identifying a rank  $r$  spike (sum of  $r$  random outer products) corrupted by Gaussian noise. Specifically, we consider the random matrix  $\boldsymbol{M} = \boldsymbol{G} + (\alpha / \sqrt{n}) \sum_{i=1}^{r} \boldsymbol{u}_i \boldsymbol{v}_i^{\mathrm{T}}$  where all the coordinates of  $\boldsymbol{G}, \boldsymbol{u}_i, \boldsymbol{v}_i$  are independently sampled from  $N(0,1)$ . We show that if there is an algorithm that uses  $k$  iterations in each round and identifies the spike with probability  $\geq 9/10$ , then it must run for  $\Omega(\log(n^2/k) / \log \log n)$  rounds by appealing to the lower bound we described above for the case of  $r = 1$ . We show that if there is an algorithm for  $r > 1$  that requires  $t < O(\log(n^2/k) / \log \log n)$  adaptive rounds, then it can be used to solve the rank 1 spike estimation problem in  $t$  rounds as well which contradicts the lower bound.
We then provide lower bounds on approximate algorithms for a host of problems such as spectral norm low rank approximation (LRA), Schatten norm LRA, Ky-Fan norm LRA and reduced rank regression, by showing that algorithms to solve these problems can be used to estimate the spike  $\boldsymbol{u}\boldsymbol{v}^{\mathsf{T}}$  in the random matrix  $G + (\alpha/\sqrt{n})\boldsymbol{u}\boldsymbol{v}^{\mathsf{T}}$ , and then use the aforementioned lower bounds on algorithms that can estimate the spike. Although our hard distribution is supported on non-symmetric matrices, we are still able to obtain lower bounds for algorithms for spectral norm LRA (rank  $r \geq 2$ ) for symmetric matrices as well by considering a suitably defined symmetric random matrix using our hard distribution. Let  $\boldsymbol{A} \coloneqq \boldsymbol{G} + (\alpha/\sqrt{n})\boldsymbol{u}\boldsymbol{v}^{\mathsf{T}}$  be the hard distribution in the case of rank  $r = 1$ .
We symmetrize the matrix by considering,  $\mathbf{A}_{\mathrm{sym}} = \left[ \begin{array}{ll}0 & \mathbf{A}\\ \mathbf{A}^{\top} & 0 \end{array} \right]$ . This symmetrization, as opposed
to  $AA^{\top}$  or  $A^{\top}A$ , has the advantage that a linear measurement of  $A_{\mathrm{sym}}$  can be obtained from an appropriately transformed linear measurement of  $A$ , thereby letting us obtain lower bounds for symmetric instances as well in the linear measurements model. However, we cannot obtain lower bounds for rank 1 spectral norm LRA for symmetric matrices using this symmetrization as the top two singular values of  $A_{\mathrm{sym}}$  are equal and hence even a zero matrix is a perfect rank 1 spectral norm LRA for  $A_{\mathrm{sym}}$  which does not give any information about the plant  $\boldsymbol{u} \otimes \boldsymbol{v}$ .
As discussed, low rank matrix recovery has been extensively studied (see [5] and references therein for earlier work). Relatedly, [13] study the Robust PCA problem where the aim is to estimate the sum of a low rank matrix and a sparse matrix from noisy vector products with the hidden matrix. [30] study the Robust PCA problem when given access to linear measurements with the hidden matrix.
For non-adaptive algorithms for low rank matrix recovery with Gaussian errors, [5] show that their selectors based on the restricted isometry property of measurement matrices are optimal up to constant factors in the minimax error sense when the noise follows a Gaussian distribution. Our Theorem 1.1 extends their lower bounds and shows that if there is any randomized measurement matrix  $M$  with  $k$  rows coupled with a recovery algorithm that outputs a reconstruction for any rank  $r$  matrix with  $\| A\|_{\mathsf{F}}^2 = \Theta (nr)$ , then it must have  $k = \Omega (n^{2 - o(1)})$ . We again note that we give lower bounds even for algorithms with multiple adaptive rounds.
Our technique to show lower bounds is to plant a low rank matrix  $(\alpha / \sqrt{n}) \sum_{i=1}^{r} u_i v_i^{\mathsf{T}}$  in an  $n \times n$  Gaussian matrix so that any "orthonormal" access to the plant is corrupted by independent Gaussian noise. Notably, this technique has been employed to obtain lower bounds on adaptive algorithms for sparse recovery in [23]. Even in the non-adaptive setting, Li and Woodruff [18] use the same hard distribution as we do to obtain sketching lower bounds for approximating Schatten  $p$  norms, the operator norm, and the Ky Fan norms. The technique to show their lower bounds is that if a sketching matrix has  $k \leq c / (r^2 s^4)$  rows<sup>3</sup>, it cannot distinguish between the random matrix  $\pmb{G}$  and the random matrix  $\pmb{G} + s \sum_{i=1}^{r} \pmb{u}_i \pmb{v}_i^{\mathsf{T}}$  where all the coordinates of  $\pmb{G}, \pmb{u}_i, \pmb{v}_i$  are drawn uniformly at random. They prove this by showing that  $d_{\mathrm{TV}}(\mathcal{L}_1, \mathcal{L}_2)$  is small if  $\mathcal{L}_1$  is the distribution of  $M \cdot \operatorname{vec}(G)$  and  $\mathcal{L}_2$  is the distribution of  $M \operatorname{vec}(G + s \sum_{i=1}^{n} \pmb{u}_i \pmb{v}_i^{\mathsf{T}})$  for any fixed measurement matrix  $M$  with  $k \leq c / (r^2 s^4)$  rows. Their techniques do not extend to the plant estimation in the distribution  $\pmb{G} + s \sum_{i=1}^{r} \pmb{u}_i \pmb{v}_i^{\mathsf{T}}$  as the statement they prove only says that over the randomness of  $\pmb{G}, \pmb{u}_i, \pmb{v}_i$ , the response distribution for  $\pmb{G} + s \sum_{i=1}^{r} \pmb{u}_i \pmb{v}_i^{\mathsf{T}}$  is close to the response distribution for  $\pmb{G}$ , but in our case, we want the distributions  $\mathcal{L}_{u,v}$  and  $\mathcal{L}_{u',v'}$  to be indistinguishable, where  $\mathcal{L}_{u,v}$  is the response distribution for  $\pmb{G} + (\alpha / \sqrt{n}) \sum_{i=1}^{n} u_i v_i^{\mathsf{T}}$ .
Later, [28] considered the distribution of the symmetric random matrix  $\boldsymbol{W} + \lambda \boldsymbol{U}\boldsymbol{U}^{\top}$ , where  $\boldsymbol{W}$  is the  $n \times n$  symmetric Wigner matrix  $(\boldsymbol{G} + \boldsymbol{G}^{\top}) / \sqrt{2}$  and  $\boldsymbol{U}$  is a uniformly random  $n \times r$  matrix with orthonormal columns. They focus on obtaining lower bounds on adaptive algorithms that estimate the spike  $\boldsymbol{U}$  in the matrix-vector product model. In particular, they show that if  $Q$  is a basis for the query space spanned by any deterministic algorithm after querying  $k$  adaptive matrix-vector queries, then  $\lambda_{r}(Q^{\top}\boldsymbol{U}\boldsymbol{U}^{\top}Q)$  grows as  $\sim \lambda^{k / r}$ . Using this, they show that any algorithm which, given access to an arbitrary symmetric matrix  $A$  in the matrix-vector product query model, must use  $\Omega (r\log (n) / \sqrt{\mathrm{gap}})$  adaptive queries to output an  $n \times r$  orthonormal matrix  $V$  satisfying, for a small enough constant  $c$ ,
$$
\langle V, A V \rangle \geq (1 - c \cdot \operatorname {g a p}) \sum_ {i = 1} ^ {r} \lambda_ {i} (A), \tag {3}
$$
where  $\mathrm{gap} = (\lambda_r(A) - \lambda_{r + 1}(A)) / \lambda_r(A)$ . We note the above guarantee is non-standard in the low rank approximation literature, which instead focuses more on approximation algorithms for Frobenius norm and spectral norm LRA. While in the matrix-vector product model, their hard distribution helps in getting lower bounds for numerical linear algebra problems on symmetric matrices, it seems that our hard distribution  $G + (\alpha /\sqrt{n})\sum_{i = 1}^{r}\pmb {u}_i\pmb {v}_i^\top$  is easier to understand in the linear measurement model and gives the important property that the noise between rounds is independent, which is
Note that we assume a measurement matrix has orthonormal rows and that each measurement is corrupted by Gaussian noise of variance 1
<sup>3</sup>Their lower bound is a bit more general but we state this formulation for simplicity.
what lets us reduce the matrix-recovery problem with noisy measurements to spike estimation in  $\pmb{G} + (\alpha / \sqrt{n}) \sum_{i=1}^{r} \pmb{u}_i \pmb{v}_i^{\mathsf{T}}$ .
Recently, Bakshi and Narayanan [1] obtained a tight lower bound for rank-1 spectral norm low rank approximation problem in the matrix-vector product model. They show that  $\Omega (\log n / \sqrt{\varepsilon})$  matrix vector products are required to obtain a  $1 + \varepsilon$  spectral norm low rank approximation. We stress that while our results are not for  $1 + \varepsilon$  approximations, they hold in the stronger linear measurements model.
Given an integer  $n \geq 1$ , we use  $[n]$  to denote the set  $\{1, \ldots, n\}$ . For a vector  $v \in \mathbb{R}^n$ ,  $\|v\|_2 := (\sum_{i \in [n]} v_i^2)^{1/2}$  denotes the Euclidean norm. For a matrix  $A \in \mathbb{R}^{n \times d}$ ,  $\|A\|_{\mathsf{F}}$  denotes the Frobenius norm  $(\sum_{i,j} A_{ij}^2)^{1/2}$  and  $\|A\|_2$  denotes the spectral norm  $\sup_{x \neq 0} \|Ax\|_2 / \|x\|_2$ . For  $p \geq 2$ , we use  $\|A\|_{\mathsf{S}_p}$  to denote the Schatten- $p$  norm  $(\sum_i \sigma_i^p)^{1/p}$  and for  $p \in [\min(n,d)]$ , we use  $\|A\|_{\mathsf{F}_p}$  to denote the Ky-Fan  $p$  norm  $\sum_{i=1}^p \sigma_i$ , where  $\sigma_1, \ldots, \sigma_{\min(n,d)}$  denote the singular values of the matrix  $A$ . Given a parameter  $k$ , the matrix  $[A]_k$  denotes the best rank  $k$  approximation for matrix  $A$  in Frobenius norm. By the Eckart-Young-Mirsky theorem, the best rank  $k$  approximation under any unitarily invariant norm, which includes the spectral norm, Frobenius norm, Schatten norms and Ky-Fan norms, is given by truncating the Singular Value Decomposition of the matrix  $A$  to top  $k$  singular values. We describe the Bayes Risk Lower Bounds that we use to prove Lemma 3.1 in the Appendix. Our presentation is based on [7]. Due to space constraints, we have placed all proofs in the appendix.
3 Number of Linear Measurements Versus Number of Adaptive Rounds
We now state the main theorem which shows a lower bound on the number of adaptive rounds required for any deterministic algorithm to estimate the plant when the input is a random matrix  $\mathbf{G} + (\alpha / \sqrt{n}) \sum_{i=1}^{r} \mathbf{u}_i \mathbf{v}_i^{\mathrm{T}}$ . We use this theorem to prove Theorem 1.1 and lower bounds for other numerical linear algebra problems.
We prove the lower bound for the rank-  $r$  plant estimation by first proving a lower bound on the rank-1 plant estimation and then reducing the rank-1 recovery problem to rank-  $r$  recovery problem. For the rank-1 recovery problem, we prove the following lemma:
Lemma 3.1. Given an integer  $n$ , and parameters  $\alpha \geq 10$  and  $\gamma \geq 1$ , define the  $n \times n$  random matrix  $M = G + s u v^{\top}$  for  $s \coloneqq \alpha / \sqrt{n}$ , where the entries of  $G, u, v$  are drawn independently from the distribution  $N(0,1)$ . Let  $Alg$  be any  $t$ -round deterministic algorithm that makes  $k \geq n$  adaptive linear measurements in each round. Let  $Q^{(j)} \in \mathbb{R}^{k \times n^2}$  denote the matrix of general linear queries made by the algorithm in round  $j$  and  $Q$  be a matrix with orthonormal rows such that  $\text{rowspan}(Q) = \text{rowspan}(Q^{(1)}) + \ldots + \text{rowspan}(Q^{(t)})$ . Then for all  $t$  such that  $O(k \log(n)) \leq (K\alpha^2\gamma^2)^t k \leq O(n^2)$  for a universal constant  $K$ ,
$$
\operatorname * {P r} _ {M = \boldsymbol {G} + s \boldsymbol {u} \boldsymbol {v} ^ {\intercal}} [ \| Q (\boldsymbol {u} \otimes \boldsymbol {v}) \| _ {2} ^ {2} \leq (3 K) k (K \alpha^ {2} \gamma^ {2}) ^ {t} ] \geq (1 - 1 / (1 0 \gamma)) ^ {t} - 1 / \operatorname {p o l y} (n).
$$
Setting  $\gamma = O(\log (n))$ , the theorem shows that if  $t \leq c \log (n^2 / k) / (\log \log (n) + \log (\alpha))$  for a small enough constant  $c$ , then for any  $t$ -round deterministic algorithm,
$$
\Pr_ {M = G + s u v ^ {\top}} [ \| Q (\boldsymbol {u} \otimes \boldsymbol {v}) \| _ {2} ^ {2} \leq n ^ {2} / 1 0 0 ] \geq 4 / 5. \tag {4}
$$
Setting  $\gamma = O(1)$ , we obtain that if  $t \leq c\log(n^2 / k) / (\log(\alpha))$  for a small enough constant  $c$ , then
$$
\Pr_ {M = G + s u v ^ {\top}} [ \| Q (\boldsymbol {u} \otimes \boldsymbol {v}) \| _ {2} ^ {2} \leq n ^ {2} / 1 0 0 ] \geq 1 / \text {p o l y} (n). \tag {5}
$$
The proof of the above lemma is in Appendix B. The above lemma directly shows lower bounds on any deterministic algorithm that can approximate the rank-1 planted matrix. We show that the lower bounds can be extended to algorithms that estimate the rank- $r$  plant as well.
Theorem 3.2. Let  $n$  and  $r \leq n/2$  be input parameters and  $\alpha$  be a large enough constant. Let the random matrix  $\mathbf{G} + (\alpha/\sqrt{n})\sum_{i=1}^{r}\mathbf{u}_i\mathbf{v}_i^\top$  be the input which can be accessed using linear measurements. If  $\mathsf{Alg}$  is a  $t$ -round adaptive algorithm that uses  $k$  linear measurements and at the end of  $t$ -rounds outputs a matrix  $\hat{A}$  such that with probability  $\geq 9/10$  over the randomness of the input and the internal randomness of the algorithm,
$$
\| \hat {A} - \sum_ {i = 1} ^ {r} \boldsymbol {u} _ {i} \boldsymbol {v} _ {i} ^ {\top} \| _ {\mathsf {F}} ^ {2} \leq c (n ^ {2} r)
$$
for a small enough constant  $c$ , then  $t = \Omega (\log (n^{2} / k) / (\log (\alpha) + \log \log n))$
Using the reduction from matrix recovery with noisy measurements to plant estimation with exact measurements that we described in the previous sections, we can now prove Theorem 1.1. The proof of Theorem 1.1 is in Appendix D.
Praneeth Kacham and David P. Woodruff acknowledge the partial support from a Simons Investigator Award.