SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Praneeth Kacham
pkacham@cs.cmu.edu
Carngie Mellon University
We study lower bounds on adaptive sensing algorithms for recovering low rank matrices using linear measurements. Given an  $n \times n$  matrix  $A$ , a general linear measurement  $S(A)$ , for an  $n \times n$  matrix  $S$ , is just the inner product of  $S$  and  $A$ , each treated as  $n^2$ -dimensional vectors. By performing as few linear measurements as possible on a rank- $r$  matrix  $A$ , we hope to construct a matrix  $\hat{A}$  that satisfies
$$
\| A - \hat {A} \| _ {\mathsf {F}} ^ {2} \leq c \| A \| _ {\mathsf {F}} ^ {2}, \tag {1}
$$
for a small constant  $c$ . Here  $\| A \|_{\mathsf{F}}$  denotes the Frobenius norm  $(\sum_{i,j} A_{i,j}^2)^{1/2}$ . It is commonly assumed that when measuring  $A$  with  $S$ , the response is corrupted with an independent Gaussian random variable of mean 0 and variance  $\sigma^2$ . Candés and Plan (IEEE Trans. Inform. Theory 2011) study non-adaptive algorithms for low rank matrix recovery using random linear measurements. They use the restricted isometry property (RIP) of Random Gaussian Matrices to give tractable algorithms to estimate  $A$  from the measurements.
At the edge of the noise level where recovery is information-theoretically feasible, it is known that their non-adaptive algorithms need to perform  $\Omega(n^2)$  measurements, which amounts to reading the entire matrix. An important question is whether adaptivity helps in decreasing the overall number of measurements. While for the related problem of sparse recovery, adaptive algorithms have been extensively studied, as far as we are aware adaptive algorithms and lower bounds on them seem largely unexplored for matrix recovery. We show that any adaptive algorithm that uses  $k$  linear measurements in each round and outputs an approximation as in (1) with probability  $\geq 9/10$  must run for  $t = \Omega(\log(n^2/k)/\log\log n)$  rounds. Our lower bound shows that any adaptive algorithm which uses  $n^{2-\beta}$  (for any constant  $\beta > 0$ ) linear measurements in each round must run for  $\Omega(\log n/\log\log n)$  rounds to compute a good reconstruction with probability  $\geq 9/10$ . Hence any adaptive algorithm that has  $o(\log n/\log\log n)$  rounds must use an overall  $\Omega(n^2)$  linear measurements. Our techniques also readily extend to obtain lower bounds on adaptive algorithms for tensor recovery.
Our hard distribution also allows us to give a measurement-vs-rounds trade-off for many sensing problems in numerical linear algebra, such as spectral norm low rank approximation, Frobenius norm low rank approximation, singular vector approximation, and more.
Sparse recovery, also known as compressed sensing, is the study of under-determined systems of equations subject to a sparsity constraint. Suppose we know that an unknown vector  $x \in \mathbb{R}^n$
has at most  $r \ll n$  non-zero entries. We would like to use a measurement matrix  $M \in \mathbb{R}^{k \times n}$  to recover the vector  $x$  given measurements  $y = Mx$ . The number  $k$  of measurements is an important parameter we would like to optimize as it models the equipment cost in physical settings and running times in computational settings. Ideally one would like for the number  $k$  of measurements to scale proportionally to the sparsity of the unknown vector  $x$ .
One way of modeling sparse recovery is as "stable sparse recovery". Here we want a distribution  $\mathcal{M}$  over  $k\times n$  matrices such that for any  $x\in \mathbb{R}^n$ , with probability  $\geq 1 - \delta$  over  $M\sim \mathcal{M}$ , we can construct a vector  $\hat{x}$  from  $Mx$  such that
$$
\| x - \hat {x} \| _ {p} \leq (1 + \varepsilon) \min  _ {r \text {- s p a r s e} x ^ {\prime}} \| x - x ^ {\prime} \| _ {p}.
$$
Note that the above formulation is more robust as it does not require the underlying vector  $x$  to be exactly  $r$ -sparse and instead asks to recover the top  $r$  coordinates of  $x$ .
For  $p = 2$ , it is known that  $k = \Theta (\varepsilon^{-1}r\log (n / r))$  measurements is both necessary and sufficient [6, 9, 22, 4]. See [23] and references therein for upper and lower bounds for other values of  $p$ .
In the same vein, the problem of low rank matrix recovery has been studied. See [5, 35] and references therein for earlier work and numerous applications. In this problem, the aim is to recover a low rank matrix  $A$  using linear measurements. Here we want a distribution  $\mathcal{M}$  over linear operators  $M: \mathbb{R}^{n \times n} \to \mathbb{R}^k$ , such that for any low rank matrix  $A$ , with probability  $\geq 1 - \delta$  over  $M \sim \mathcal{M}$ , we can construct a matrix  $\hat{A}$  from  $\mathcal{M}(A)$  such that the reconstruction error  $\| A - \hat{A} \|_{\mathsf{F}}^2$  is small. Note that a linear operator  $M: \mathbb{R}^{n \times n} \to \mathbb{R}^k$  can be equivalently represented as a matrix  $M \in \mathbb{R}^{k \times n^2}$  such that  $M \cdot \operatorname{vec}(A) = M(A)$  where  $\operatorname{vec}(A)$  denotes the appropriate flattening of the matrix  $A$  into a vector. We call the rows of  $M$  linear measurements. Without loss of generality, we can assume that the rows of the matrix  $M$  are orthonormal, as the responses for non-orthonormal queries can be obtained via a simple change of basis.
In addition, to model the measurement error that occurs in practice, it is a standard assumption that when querying with  $M$ , we receive  $M \cdot \operatorname{vec}(A) + g$ , where  $g$  is a  $k$ -dimensional vector with independent Gaussian random variables of mean 0 and variance  $\sigma^2$ , and we hope to reconstruct  $A$  with small error from  $M \cdot \operatorname{vec}(A) + g$ . Clearly, when  $A$  has rank  $r$ , we need to perform  $\Omega(nr)$  linear measurements, as the matrix  $A$  has  $\Theta(nr)$  independent parameters. Hence, the aim is to perform not many more linear measurements than  $nr$  while being able to obtain an estimate  $\hat{A}$  for  $A$  with low estimation error.
Given a rank- $r$  matrix  $A$ , [5] show that if the  $k \times n^2$  matrix  $M$  is constructed with independent standard Gaussian entries, then with probability  $\geq 1 - \exp(-cn)$ , an estimate  $\hat{A}$  can be constructed from  $M \cdot \operatorname{vec}(A) + g$  such that  $\|A - \hat{A}\|_{\mathbb{F}}^2 \leq O(nr\sigma^2 / k)$ . They use the restricted isometry property (RIP) of the Gaussian matrix  $M$  to obtain algorithms that give an estimate  $\hat{A}$ . The error bound is intuitive since the reconstruction error increases with increasing noise and proportionally goes down when the number of measurements  $k$  increases.
While we formulated the sparse recovery and matrix recovery problems in a non-adaptive way, there have been works which study adaptive algorithms for sparse recovery. Here we can produce matrices  $M^{(i)}$  adaptively based on the responses received  $M^{(1)}x$ ,  $M^{(2)}x$ , ...,  $M^{(i-1)}x$  and the hope is that allowing for adaptive algorithms with a small number of adaptive rounds, we obtain algorithms that overall perform fewer linear measurements than non-adaptive algorithms with the same reconstruction error. It is additionally assumed that the noise across different rounds is independent. For sparse recovery, in the case of  $p = 2$ , it is known that over  $O(\log \log n)$  rounds adaptivity, a total of  $O(\varepsilon^{-1}r\log \log (n\log (n / r)))$  linear measurements suffices [14, 21], improving over the requirement of  $\Theta (\varepsilon^{-1}r\log (n / r))$  linear measurements for non-adaptive algorithms. For a constant  $\varepsilon$ , for any number of rounds, it is known that  $\Omega (r + \log \log n)$  linear measurements are required [23]. Recently, [16] gave a lower bound on the total number of measurements if the number of adaptive rounds is constant.
While there has been a lot of interest in adaptive sparse recovery, both from the algorithms and the lower bounds perspective, the adaptive matrix recovery problem surprisingly does not seem to have any lower bounds. In adaptive matrix recovery, similar to adaptive sparse recovery, one is allowed to query a matrix  $M^{(i)}$  in round  $i$  based on the responses received in the previous rounds, and again the hope is that with more rounds of adaptivity, the total number of linear measurements that is to
be performed decreases. There is some work that studies adaptive matrix recovery with 2 rounds of adaptivity (see [34] and references therein) but the full landscape of adaptive algorithms for matrix recovery seems unexplored.
We address this from the lower bounds side in this work. We show lower bounds on adaptive algorithms that recover a rank- $r$  matrix of the form  $(\alpha/\sqrt{n}) \sum_{i=1}^{r} \boldsymbol{u}_i \boldsymbol{v}_i^{\mathsf{T}}$ , where the coordinates of  $\boldsymbol{u}_i$  and  $\boldsymbol{v}_i$  are sampled independently from the standard Gaussian distribution. Without loss of generality, we assume that the measurement matrix  $M^{(i)}$  in each round  $i$  has orthonormal rows. We also assume that for  $i \neq j$ , the measurement matrices  $M^{(i)}(M^{(j)})^{\mathsf{T}} = 0$ , i.e., measurements across adaptive rounds are orthonormal, since non-orthonormal measurements can be reconstructed from orthonormal measurement matrices by a change of basis.
We now give an alternate way of looking at the adaptive sparse recovery problem: Fix a set of vectors  $u_{1},\ldots ,u_{r}$  and  $v_{1},\ldots ,v_{r}$  and let the underlying matrix we want to reconstruct be  $(\alpha /\sqrt{n})\sum_{i = 1}^{r}u_{i}v_{i}^{\mathsf{T}}$  for a large enough constant  $\alpha$ . In round 1, we query a matrix  $M^{(1)}\in \mathbb{R}^{k\times n^2}$  drawn from an appropriate distribution and receive the response vector  $r^{(1)} = M^{(1)}\mathrm{vec}((\alpha /\sqrt{n})\sum_{i = 1}^{r}u_{i}v_{i}^{\mathsf{T}}) + g^{(1)}$  where  $g^{(1)}$  is a vector of independent Gaussian random variables with mean 0 and variance  $\sigma^2$ . In round 2, as a function of  $M^{(1)}$  and  $r^{(1)}$  and our randomness, we query a random matrix  $M^{(2)}[r^{(1)}]\in \mathbb{R}^{k\times n^2}$  and receive a response vector  $r^{(2)} = (M^{(2)}[r^{(1)}])\mathrm{vec}((\alpha /\sqrt{n})\sum_{i = 1}^{r}u_{i}v_{i}^{\mathsf{T}}) + g^{(2)}$  where  $g^{(2)}$  is again a vector with independent Gaussian entries and independent of  $g^{(1)}$ , and so on. Crucially, using the rotational invariance of Gaussian random vectors, if  $G$  is an  $n\times n$  matrix with independent Gaussian random variables with mean 0 and variance  $\sigma^2$ , the response  $r^{(1)}$  has the same distribution as  $(M^{(1)})\cdot \mathrm{vec}((\alpha /\sqrt{n})\sum_{i = 1}^{r}u_{i}v_{i}^{\mathsf{T}} + G)$  and as  $M^{(2)}[r^{(1)}]$  is chosen to be orthonormal to  $M^{(1)}$ , the distribution of  $r^{(2)}$  conditioned on  $r^{(1)}$  is the same as that of  $(M^{(2)}[r^{(1)}])\cdot (\mathrm{vec}((\alpha /\sqrt{n})\sum_{i = 1}^{r}u_{i}v_{i}^{\mathsf{T}} + G))$ .
Thus, any adaptive matrix recovery algorithm can be seen as performing non-noisy adaptive measurements on the matrix  $(\alpha / \sqrt{n}) \sum_{i=1}^{r} u_i v_i^{\mathsf{T}} + G$  where the Gaussian matrix  $G$  is sampled independently of the measurement algorithm. From the responses the algorithm receives, it then tries to reconstruct the matrix  $(\alpha / \sqrt{n}) \sum_{i=1}^{r} u_i v_i^{\mathsf{T}}$ . This way of looking at adaptive sparse recovery immediately yields an adaptive algorithm: when the smallest singular value of  $(\alpha / \sqrt{n}) \sum_{i=1}^{r} u_i v_i^{\mathsf{T}}$  is a constant times larger than  $\| G \|_2$ , then the power method with a block size of  $r$  outputs an approximation of  $(\alpha / \sqrt{n}) \sum_{i=1}^{r} u_i v_i^{\mathsf{T}}$  in  $O(\log n)$  rounds. Note that  $r$  matrix-vector products can be implemented using  $nr$  linear measurements. More generally, Gu [12] showed that using power iteration with a block size of  $k / n$  (for  $k \geq nr$ ), we can obtain an approximation in  $O(\log (n^2 / k))$  adaptive rounds. Thus the already existing algorithms exhibit a no. of measurements vs no. of rounds trade-off.
From results in random matrix theory, we have that  $\| G\| _2\approx \sigma \sqrt{n}$  with high probability. And as we are interested in reconstruction when the vectors  $u_{1},\ldots ,u_{r}$  and  $v_{1},\ldots ,v_{r}$  follow the Gaussian distribution we also have that  $\| u_i\| _2\approx \| v_i\| _2\approx \sqrt{n}$  simultaneously with large probability. Thus, to make the extraction of  $(\alpha /\sqrt{n})\sum_{i = 1}^{r}u_{i}v_{i}^{\mathsf{T}}$  information-theoretically possible, we need to assume that  $\alpha \gtrsim \sigma$ . Hence, in this work we assume that  $\sigma = 1$  and that  $\alpha$  is a large enough constant, and we study lower bounds on adaptive matrix recovery algorithms.
If the vectors  $u_{1}, \ldots, u_{r}$  and  $v_{1}, \ldots, v_{r}$  are sampled from the standard  $n$  dimensional Gaussian distribution and  $r \leq n/2$ , we also have that with high probability  $\| (\alpha / \sqrt{n}) \sum_{i=1}^{r} u_{i} v_{i}^{\top} \|_{\mathbb{F}}^{2} \approx \alpha^{2} n r$ . Now the algorithms of [5], which use a uniform random Gaussian matrix  $M$  with  $m$  rows as the measurement matrix, for  $(\alpha / \sqrt{n}) \sum_{i=1}^{r} u_{i} v_{i}^{\top}$  reconstruct a matrix  $\hat{A}$  such that  $\| \hat{A} - (\alpha / \sqrt{n}) \sum_{i=1}^{r} u_{i} v_{i}^{\top} \|_{\mathbb{F}}^{2} \leq C \frac{nr \sigma_{m}^{2}}{m}$  where  $\sigma_{m}$  is the measurement error. As we assumed above that  $\sigma = 1$  when measuring with a matrix with orthonormal rows, we assume that the measurement error  $\sigma_{m}$  when measuring with a Gaussian matrix is  $n$ , as each row of  $M$  has  $n^{2}$  independent Gaussian coordinates and therefore has a norm  $\approx n$ . Thus, using reconstruction algorithms from [5], we obtain a matrix  $\hat{A}$  satisfying  $\| \hat{A} - (\alpha / \sqrt{n}) \sum_{i=1}^{r} u_{i} v_{i}^{\top} \|_{\mathbb{F}}^{2} \leq C \frac{n^{3} r}{m}$ . Now, to make  $\| \hat{A} - (\alpha / \sqrt{n}) \sum_{i=1}^{r} u_{i} v_{i}^{\top} \|_{\mathbb{F}}^{2} \leq (1/10) \| (\alpha / \sqrt{n}) \sum_{i=1}^{r} u_{i} v_{i}^{\top} \|_{\mathbb{F}}^{2}$ , we need to set  $m = \Theta(n^{2}/\alpha^{2})$ . Hence, in the parameter regimes we
<sup>1</sup>In general, in the sparse recovery and matrix recovery models, the algorithms may make the same query and obtain responses with independent noise added which can be used to say obtain more accurate result using the median/mean estimator. But all the algorithms we are aware of for sparse recovery and matrix recovery do not explore independence of noise across queries in this way.
study, the algorithms of [5] need to perform  $\Omega(n^2)$  non-adaptive queries, which essentially means that they have to read a constant fraction of the  $n^2$  entries in the matrix. While the power method performs  $O(nr)$  linear measurements in each round over  $O(\log n)$  adaptive rounds to output an approximation  $\hat{A}$ . The question we ask is:
"Is the power method optimal? Are there algorithms that have  $o(\log n)$  adaptive rounds and use  $n^{2 - \beta}$  measurements in each round?"
We answer this question by showing that any algorithm that has  $o(\log n / \log \log n)$  adaptive rounds must use  $\geq n^{2 - \beta}$  linear measurements, for any constant  $\beta > 0$ , in each round. The lower bound shows that power method is essentially optimal if we want to use  $n^{2 - \beta}$  linear measurements in each round and that any algorithm with  $o(\log n / \log \log n)$  adaptive rounds essentially reads the whole matrix.
We further obtain a rounds vs measurements trade-off for many numerical linear algebra problems in the sensing model. In this model, there is an  $n \times n$  matrix  $A$  with which we can interact only using general linear measurements and we want to solve problems such as spectral norm low rank approximation of  $A$ , Frobenius norm low rank approximation of  $A$ , etc. In general, numerical linear algebra algorithms assume that they either have access to the entire matrix or that the matrix is accessible in the matrix-vector product model where one can query a vector  $v$  and obtain the result  $A \cdot v$ . Recently, the vector-matrix-vector product model has received significant attention as well. Linear measurements are more powerful than both the matrix-vector product model and the vector-matrix-vector product model. Any matrix vector product  $A \cdot v$  can be computed using  $n$  linear measurements of  $A$  and any vector-matrix-vector product  $u^{\mathrm{T}}Av$  can be computed using a single linear measurement of  $A$ . Thus, the model of general linear measurements may lead to faster algorithms.
For certain problems in numerical linear algebra, general linear measurements are significantly more powerful than the matrix-vector product model. Indeed, for computing the trace of an  $n \times n$  matrix  $A$ , one can do this exactly with a single deterministic general linear measurement, just by adding up the diagonal entries of  $A$ . However, in the matrix-vector product model, it is known that  $\Omega(n)$  matrix-vector products are needed to compute the trace exactly [29], even if one uses randomization. A number of problems were studied in the vector-matrix-vector product model in [24], and in [32] it was shown that to approximate the trace of  $A$  up to a  $(1 + \varepsilon)$ -factor with probability  $1 - \delta$ , one needs  $\Omega(\varepsilon^{-2} \log(1/\delta))$  queries. This contrasts sharply with the single deterministic general linear measurement for computing the trace exactly. Thus, there are good reasons to conjecture that general linear measurements may lead to algorithms requiring fewer rounds of adaptivity compared to algorithms in the matrix-vector product query model. Surprisingly, our lower bounds show that for many numerical linear algebra problems, linear measurements do not give much advantage over matrix-vector products.
We assume that there is an unknown rank- $r$  matrix  $A \in \mathbb{R}^{n \times n}$  to be recovered. Given any linear measurement  $q \in \mathbb{R}^{n^2}$ , we receive a response  $\langle q, \operatorname{vec}(A) \rangle + g$ , where  $g \sim N(0, \|q\|_2^2)$ . We further assume that the noise for two different measurements is independent. Without loss of generality, we also assume throughout that all the queries an algorithm makes across all rounds form a matrix with orthonormal rows. Our main result for adaptive matrix sensing is as follows:
Theorem 1.1. There exists a constant  $c$  such that any randomized algorithm which makes  $k \geq nr$  noisy linear measurements of an arbitrary rank- $r$  matrix  $A$  with  $\| A \|_{\mathsf{F}}^2 = \Theta (nr)$  in each of  $t$  rounds, and outputs an estimate  $\hat{A}$  satisfying  $\| A - \hat{A} \|_{\mathsf{F}}^2 \leq c \| A \|_{\mathsf{F}}^2$  with probability  $\geq 9/10$  over the randomness of the algorithm and the Gaussian noise, requires  $t = \Omega (\log (n^2 / k) / (\log \log n))$ .
Dependence on noise In our results, we assumed that given a linear measurement  $q \in \mathbb{R}^{n^2}$ , the response is distributed as  $N(\langle q, \mathrm{vec}(A) \rangle, \|q\|_2^2)$ . Our lower bounds also hold when the response is distributed as  $N(\langle q, \mathrm{vec}(A) \rangle, \sigma^2 \|q\|_2^2)$  for any parameter  $\sigma$  such that  $c' \leq \sigma \leq 1$ , where  $c'$  is a constant. This can be seen by simply scaling the matrix  $A$  in the theorem above and adjusting the constants while proving the theorem.
Application	Failure Probability	Lower Bound
2-approximate spectral LRA	0.1	c log(n2/k)/ log log n
2-approximate spectral LRA	1/poly(n)	c log(n2/k)
2-approximate Schatten p LRA	0.1	c log(n2/k)/(1/p) log(n)+log log n
2-approximate Ky-Fan p LRA	0.1	c log(n2/k)/ log(p)+log log n
1 + 1/n-approximate Frobenius LRA	0.1	c log(n2/k)/ log log n
0.1-approximate i-th singular vectors	0.1	c log(n2/k)/ log log n
2-approximate spectral reduced rank regression	0.1	c log(n2/k)/ log log n
Table 1: Number of rounds lower bound for algorithms using  $k$  general linear measurements in each round. Our bound for 2-approximate spectral LRA algorithm with constant failure probability is optimal up to a log log  $n$  factor, and our 2-approximate spectral low rank approximation (LRA) lower bound for algorithms with failure probability  $1 / \operatorname{poly}(n)$  is optimal up to a constant factor.
Tensor Recovery The problem of tensor recovery with linear measurements has also been studied (see [25, 11] and references therein) where given a low rank tensor, the task is to recover an approximation to the tensor with few linear measurements. Our techniques can also be used to obtain lower bounds on adaptive algorithms for tensor recovery. Our main tool, Lemma 3.1, readily extends to tensors of higher orders by using the corresponding tail bounds from [31].
Numerical Linear Algebra We also derive lower bounds for many numerical linear algebra problems in the linear measurements model. Table 1 shows our lower bounds on the number of adaptive rounds required for any randomized algorithm using  $k$  general linear measurements in each round. See Appendix E for precise statements and proofs.
Throughout the paper, we use uppercase letters such as  $G$ ,  $M$ ,  $Q$  to denote matrices and lowercase letters such as  $u$ ,  $v$  to denote vectors. For vectors  $u$ ,  $v \in \mathbb{R}^n$ ,  $u \otimes v \in \mathbb{R}^{n^2}$  denotes the tensor product of  $u$  and  $v$ . For an arbitrary matrix  $M \in \mathbb{R}^{m \times n}$ , the vector  $\operatorname{vec}(M) \in \mathbb{R}^{mn}$  denotes a flattening of the matrix  $M$  with the convention that  $\operatorname{vec}(uv^{\mathrm{T}}) = u \otimes v$ . We use boldface symbols such as  $M$ ,  $G$ ,  $u$ ,  $v$  to denote that these objects are explicitly sampled from an appropriate distribution. We use  $\pmb{g}_k$  to denote a multivariate Gaussian in  $k$  dimensions where each coordinate is independently sampled from  $N(0,1)$ . We also use  $G^{(j)}$  to denote a collection of  $j$  independent multivariate Gaussian random variables of appropriate dimensions.
1.3 Our Techniques
Using the rotational invariance of the Gaussian distribution, we argued that any adaptive randomized low rank matrix recovery algorithm with access to a hidden matrix  $\left( \frac{\alpha}{\sqrt{n}} \right) \sum_{i=1}^{r} u_i v_i^{\mathsf{T}}$  using noisy linear measurements can be seen as a randomized algorithm that has access to a random matrix  $\left( \frac{\alpha}{\sqrt{n}} \right) \sum_{i=1}^{r} u_i v_i^{\mathsf{T}} + G$  using perfect linear measurements where each coordinate of  $G$  is independently sampled from a Gaussian distribution.
Let  $\mathcal{A}$  be any algorithm that satisfies the matrix recovery guarantees with, say, a success probability  $\geq 9 / 10$ . Let  $\mathcal{A}(X,\sigma ,\gamma)$  be the output of the randomized algorithm  $\mathcal{A}$ , where the hidden matrix is  $X$ , the random seed of  $\mathcal{A}$  is denoted by  $\sigma$ , and  $\gamma$  denotes the measurement randomness. We have that if  $X$  has rank  $r$  and satisfies  $\| X\|_{\mathsf{F}}^{2} = \Theta (nr)$ , then
$$
\Pr_ {\boldsymbol {\sigma}, \boldsymbol {\gamma}} [ \mathcal {A} (X, \boldsymbol {\sigma}, \boldsymbol {\gamma}) \text {i s c o r r e c t} ] \geq 9 / 1 0.
$$
We say  $\mathcal{A}(X,\sigma ,\gamma)$  is correct when the output  $\hat{X}$  satisfies  $\| \hat{X} -X\|_{\mathsf{F}}^{2}\leq (1 / 100)\| X\|_{\mathsf{F}}^{2}$ . By the above reduction, from  $\mathcal{A}$  we have a randomized algorithm  $\mathcal{A}'$  that runs on the random matrix  $X + G$  with access to exact linear measurements and outputs a correct reconstruction  $\hat{X}$  with probability  $\geq 9 / 10$  if  $X$  has rank  $r$  and  $\| X\|_{\mathsf{F}}^{2} = \Theta (nr)$ . Thus, for all such  $X$ ,
$$
\operatorname * {P r} _ {\boldsymbol {G}, \boldsymbol {\sigma}} [ \mathcal {A} ^ {\prime} (\boldsymbol {X} + \boldsymbol {G}, \boldsymbol {\sigma}) \text {i s c o r r e c t} ] \geq 9 / 1 0.
$$