SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Each first-order oracle call has the same time complexity as one forward pass and may be computed efficiently in batches using standard deep learning frameworks [62, 1, 13]. As  $n$  is finite,  $\hat{H}_n^\circ (x_0,f,\Sigma)$  is not guaranteed to be symmetric in practice. Thus, we symmetrize our estimator in Eq. (8b). A straightforward consequence of Proposition 1 is that  $\lim_{n\to \infty}\hat{H}_n(x_0,f,\Sigma) = \nabla_x^2 h_{f,\Sigma}(x_0)$ , which we formally show in the Proof of Theorem 1 in App. C.
Estimation of SmoothGrad may be amortized with SmoothHess, obtaining  $\nabla_{x}h_{f,\Sigma}(x_0)$  at a significantly reduced cost. The main computational expense when estimating SmoothGrad is the querying of the first-order oracle. As this querying is part of SmoothHess estimation, the gradients which we compute may be averaged at a minimal additional cost of  $\mathcal{O}(nd)$  to obtain a SmoothGrad estimate. Likewise, SmoothHess may be obtained at a reduced cost of  $\mathcal{O}(nd^2)$ , the cost of the outer products, during SmoothGrad estimation. For details of our algorithm, see App. D.
Last, we prove non-asymptotic bounds for our SmoothHess estimator:
Theorem 1. Let  $f: \mathbb{R}^d \to \mathbb{R}$  be a piece-wise linear function over a finite partition of  $\mathbb{R}^d$ . Let  $x_0 \in \mathbb{R}^d$ , and denote  $\{\delta_i\}_{i=1}^n$ , a set of  $n$  i.i.d random vectors in  $\mathbb{R}^d$  distributed from  $\delta_i \sim \mathcal{N}(0, \Sigma)$ . Given  $\hat{H}_n(x_0, f, \Sigma)$  as in Eq. (8), for any fixed  $\varepsilon, \gamma \in (0,1]$ , given  $n \geq \frac{4}{\varepsilon^2} \left[ \max((C^+ \sqrt{d} + \sqrt{\frac{1}{c^+} \log \frac{4}{\gamma}})^2, (C^- \sqrt{d} + \sqrt{\frac{1}{c^-} \log \frac{4}{\gamma}})^2) \right]$  then
$$
\mathbb {P} \left(\left\| \hat {H} _ {n} - H \right\| _ {2} > \varepsilon\right) \leq \gamma , \tag {9}
$$
where  $H = \nabla_{x}^{2}[(f * q_{\Sigma})(x_{0})]$ ,  $C^{+}, C^{-}, c^{+}, c^{-} > 0$  are constants depending on the function  $f$  and covariance  $\Sigma$  and  $q_{\Sigma}: \mathbb{R}^{d} \to \mathbb{R}$  is the density function of  $\mathcal{N}(0, \Sigma)$ .
We elect to use the non-asymptotic bounds presented in Theorem 3.39 of Vershynin [87], which hold for sums of outer products of a sub-gaussian random vector with itself. This poses a challenge as  $\hat{H}_n$  is the sum of outer products of two sub-gaussian random vectors which are not necessarily equal. To deal with this issue, we separately prove non-asymptotic bounds for the outer products
*All ReLU network outputs, internal neurons, and SoftMax probabilities are Lipschitz continuous [29, 26].
Dataset	MNIST	FMNIST	CIFAR10
Function	Class Logit (↓)	Int. Neuron (↓)	Class Logit (↓)	Int. Neuron (↓)	Class Logit (↓)	Int. Neuron (↓)
ε	0.25	0.50	1.00	0.25	0.50	1.00	0.25	0.50	1.00	0.25	0.50	1.00
SH+SG (Us)	9.6e-7	7.8-6	6.7e-5	4.9e-8	4.0e-7	3.3e-6	6.5e-7	4.0e-6	4.3e-5	2.0e-8	1.8e-7	1.6e-6
SG [76]	4.5e-6	4.1e-5	3.9e-4	2.1e-7	1.7e-6	1.5e-5	3.0e-6	2.7e-5	2.6e-4	1.0e-7	9.0e-7	7.0e-6
SP (H + G)	1.2e-6	9.6e-6	8.1e-5	5.5e-8	4.4e-7	3.7e-6	9.6e-7	7.5e-6	6.5e-5	3.0e-8	2.1e-7	1.8e-6
SP G	4.6e-6	4.1e-5	3.9e-4	2.1e-7	1.7e-6	1.5e-5	3.2e-6	2.8e-5	2.6e-4	1.0e-7	8.0e-7	5.2e-6
G [73]	4.2e-3	1.7e-2	6.7e-2	2.0e-3	7.0e-3	2.9e-2	3.8e-3	1.5e-2	6.0e-2	1.0e-4	4.0e-4	1.8e-3
3.0e-1	1.2e-0
Table 1: Average  $\mathcal{P}_{MSE}$  at three radii  $\epsilon$ . Results are calculated for SmoothHess + SmoothGrad (SH+SG,Us) SmoothGrad (SG) SoftPlus Gradient (SP G) SoftPlus Hessian + SoftPlus Gradient (SP (H + G)) and the vanilla Gradient (G). Results are provided for the predicted class logit, and the penultimate neuron maximally activated by the "three," dress and cat classes for MNIST, FMNIST and CIFAR10 respectively. While SP (H + G) and SP G results are reported using the best  $\beta$  chosen from over 100 values based on validation set performance, only 3 values were checked for SH + SG and SG based upon  $\sigma = \epsilon \sqrt{\bar{d}}$ . SH + SG outperforms the competing methods for all 18 permutations of dataset, function and  $\varepsilon$ .
of the positive and negative eigenvectors of the summands. This is accomplished using an identity expressing the eigenvalues of the summands in terms of  $\Sigma^{-1}\delta$  and  $\nabla_{x}f(x_{0} + \delta)$ , which we state and prove in App. C. The proof is completed by applying the triangle inequality and union bound to combine the two bounds into Eq. (9). Our proof is included in App. C.
Datasets and Models: Experiments were conducted on a real-world spirometry regression dataset, three image datasets (MNIST [45], FMNIST [90] and CIFAR10 [43]), and one synthetic dataset (Four Quadrant). The Four Quadrant dataset consists of points  $x' \in \mathbb{R}^2$  sampled uniformly from the grid  $[-2, 2] \times [2, 2] \subseteq \mathbb{R}^2$ , with a spacing of 0.008. The label of a point  $y(x') = Kx_1'x_2' \in \mathbb{R}$  is chosen based upon its quadrant, with  $K = 5, 3, 12, -10$  for Quadrants 1, 2, 3 and 4 respectively. We train a 5-layer network on MNIST and FMNIST a ResNet-18 [32] on CIFAR10 and a 6-layer network on Four Quadrant. A 10-layer 1-D convolutional neural network was trained on the spirometry regression dataset. Additional model, hyperparameter and dataset details are outlined in App. E.
Hardware: Experiments were performed on an internal cluster using NVIDIA A100 GPUs and AMD EPYC223 7302 16-Core processors.
Metrics: Gradient-based explainers are evaluated using a Taylor expansion-like function as a proxy. We estimate a gradient Hessian pair  $\tilde{G}_{(x_0,g)}\in \mathbb{R}^d,\tilde{H}_{(x_0,g)}\in \mathbb{R}^{d\times d}$  for some function  $g:\mathbb{R}^d\to \mathbb{R}$  around point  $x_0$ . Here  $g$  is either a smooth-surrogate of  $f$  or  $f$  itself. We define the following Taylor expansion-like function:
$$
\tilde {f} _ {x _ {0}, \tilde {G} _ {g}, \tilde {H} _ {g}} (\Delta) = f (x _ {0}) + \tilde {G} _ {g} ^ {T} (\Delta - x _ {0}) + \frac {1}{2} (\Delta - x _ {0}) ^ {T} \tilde {H} _ {g} (\Delta - x _ {0}). \tag {10}
$$
For readability, we remove the dependence on  $x_0$  from  $\tilde{G}_{(x_0,g)}$  and  $\tilde{H}_{(x_0,g)}$ . Eq. (10) is almost equivalent to the Taylor-expansion of  $g$  at  $x_0$ , with the key difference that the zero-th order term is  $f(x_0)$  as opposed to  $g(x_0)$ . This is done to isolate the impact of the explainers  $\tilde{G}_g$  and  $\tilde{H}_g$ . Setting  $\tilde{H} = 0$  yields  $\tilde{f}_{x_0,\tilde{G}_g,0}(\Delta)$ , a first-order Taylor expansion-like function. For brevity, we refer to  $\tilde{f}_{x_0,\tilde{G}_g,\tilde{H}_g}$  simply as a Taylor expansion below. We use the following two metrics to quantify the efficacy of SmoothHess:
Perturbation MSE: We introduce the Perturbation Mean-Squared-Error  $(\mathcal{P}_{MSE})$  to assess the ability of  $\tilde{H}$  and/or  $\tilde{G}$  to capture the behaviour of  $f$  over a given neighborhood. Given point  $x_0$  and radius  $\varepsilon > 0$ , the  $\mathcal{P}_{MSE}$  directly measures the fidelity of  $\tilde{f}_{x_0, \tilde{G}_g, \tilde{H}_g}$  to  $f$  when restricted to the ball  $B_{\varepsilon}(x_0)$ :
$$
\mathcal {P} _ {M S E} \left(x _ {0}, f, \tilde {f} _ {x _ {0}, \tilde {G} _ {g}, \tilde {H} _ {g}}, \varepsilon\right) = \frac {1}{\operatorname {V o l} \left(B _ {\varepsilon} \left(x _ {0}\right)\right)} \int_ {x ^ {\prime} \in B _ {\varepsilon} \left(x _ {0}\right)} \left(\tilde {f} _ {x _ {0}, \tilde {G} _ {g}, \tilde {H} _ {g}} \left(x ^ {\prime}\right) - f \left(x ^ {\prime}\right)\right) ^ {2} d x ^ {\prime} \tag {11}
$$
A low  $\mathcal{P}_{MSE}$  value indicates that  $\tilde{f}_{x_0,\tilde{G}_g,\tilde{H}_g}$  is a good fit to  $f$  when restricted to  $B_{\varepsilon}(x_0)$ , and thus that  $\tilde{G}_g$  and  $\tilde{H}_g$  capture the behaviour of  $f$  over  $B_{\varepsilon}(x_0)$ .  $\mathcal{P}_{MSE}$  may be estimated for a point
Dataset	MNIST	FMNIST	CIFAR10
Attack Magnitude ε	0.25	0.50	0.75	1.25	1.75	0.25	0.50	0.75	1.25	1.75	0.1	0.2	0.3	0.4	1.0
SH+SG (Us)	93.0	80.3	48.0	10.5	2.0	79.5	46.8	25.0	3.5	0.0	62.5	38.5	26.5	15.0	4.5
SG [76]	93.3	81.8	48.8	11.3	2.8	79.5	49.3	26.3	4.0	0.0	65.0	42.0	27.5	17.0	0.0
SP (H + G)	93.0	81.8	51.5	15.8	7.5	79.8	51.0	27.5	5.3	0.8	64.5	42.0	31.0	23.5	7.5
SP G	93.3	82.3	53.8	16.3	5.0	79.8	51.5	29.5	7.8	1.0	66.5	47.5	36.0	29.5	8.5
G [73]	93.3	82.8	56.0	18.5	8.8	80.3	52.3	31.8	11.0	2.5	69.0	51.5	41.0	34.0	21.5
Random	99.8	99.5	99.0	99.0	98.8	99.3	98.0	97.3	95.5	93.8	100.0	99.5	99.0	98.5	96.5
Table 2: Post-hoc accuracy of adversarial attacks performed on the predicted SoftMax probability, at five attack magnitudes  $\epsilon$ . Lower is better. Results for SmoothHess + SmoothGrad (SH + SG, Ours), SmoothGrad (SG), SoftPlus Gradient (SP G) and SoftPlus Hessian + SoftPlus Gradient (SP (H + G)) are reported using parameters  $\Sigma = \sigma^2 I \in \mathbb{R}^{d \times d}$  and  $\beta > 0$  chosen based upon performance on a held-out validation set. We additionally compare with the vanilla (unsmoothed) Gradient (G). First order attack vectors are constructed by scaling the normalized gradient by  $\epsilon$  and subtracting from the input. Second order attack vectors are found by minimizing the corresponding second-order Taylor expansions.
$x_0$  by sampling a set of  $n \in \mathbb{N}$  points  $\{x_i'\}_{i=1}^n$  uniformly from  $B_{\varepsilon}(x_0)$ , computing the errors and MC-averaging:  $\frac{1}{n} \sum_{i=1}^{n} (\tilde{f}_{x_0, \tilde{G}_g, \tilde{H}_g}(x_i') - f(x_i'))^2 \approx \mathcal{P}_{MSE}(x_0, f, \tilde{f}_{x_0, \tilde{G}_g, \tilde{H}_g}, \varepsilon)$ .
Adversarial Attacks: Given  $\tilde{G}$  and  $\tilde{H}$ , one can use  $\tilde{f}_{x_0,\tilde{G},\tilde{H}}$  to construct adversarial attacks of any desired magnitude  $\varepsilon > 0$ . We denote  $\Delta^{*} \in \mathbb{R}^{d}$  as the perturbed input, and corresponding attack vectors by  $\Delta^{*} - x_{0}$ . Given only  $\tilde{G}_g$  (i.e.  $\tilde{H}_g = 0$ ), the first-order Taylor expansion attack yields:  $\Delta^{*} - x_{0} = -\varepsilon \tilde{G}_{g} \backslash \| \tilde{G}_{g}\|_{2}$ . Otherwise, the attack may be framed as the solution to a quadratic optimization minimizing the second-order Taylor expansion:
$$
\min  _ {\Delta \in \mathbb {R} ^ {d}} \tilde {f} _ {x _ {0}, \tilde {G} _ {g}, \tilde {H} _ {g}} (\Delta), \quad s. t. \quad \| \Delta - x _ {0} \| _ {2} \leq \varepsilon . \tag {12}
$$
Although this optimization is non-convex (as  $\tilde{H}_g$  is not guaranteed to be positive-semi-definite), it can be solved exactly [12]. For implementation details, and a discussion of the similarities and differences with Singla et al. [74], see App. D. We set  $g$  to be the smoothed version of the predicted class SoftMax probability function. Given a set of test points  $\{x_i\}_{i=1}^n$ ,  $n \in \mathbb{N}$ , we validate the efficacy of our attacks using the average post-hoc accuracy metric  $\frac{1}{n} \sum_{i=1}^{n} \mathbb{I}[\arg\max_t F(x_i) = \arg\max_t F(\Delta_i^*)]$ , where  $\mathbb{I}$  denotes the indicator function, and  $\Delta_i^*$  the output after  $x_i$  is attacked.
Setup: We provide the details for our experiments below:
Four Quadrant: We train the network to memorize the Four Quadrant dataset. We measure the interactions at  $x_0 = (0,0)^T$  using both SmoothHess and SoftPlus Hessian, estimated with granularly sampled  $\sigma^2 \in \{1e-3,\dots,1\}$  and  $\beta \in \{1e-1,\dots,4\}$ .
$\mathcal{P}_{MSE}$ : We set  $g$  to be the smoothed version of a predicted class logit or a penultimate neuron. The penultimate neuron is chosen to be the maximally activated neuron on average over the train data by the "three," dress and cat classes for MNIST, FMNIST and CIFAR10, respectively. We compute  $\mathcal{P}_{MSE}$  for three neighborhood sizes  $\varepsilon \in \{0.25, 0.50, 1.0\}$ . While over one-hundred values of  $\beta$  are checked on a validation set before selection, only three values of  $\sigma$  are checked for SmoothHess and SmoothGrad, based on the common-sense criterion  $\sigma = \varepsilon / \sqrt{d} : \sigma \in \{\varepsilon / 2\sqrt{d}, 3\varepsilon / 4\sqrt{d}, \varepsilon / \sqrt{d}\}$  outlined in Sec 4. The standard deviation of  $\mathcal{P}_{MSE}$  results is reported in App. F.
Adversarial Attacks: Adversarial attacks are performed after selecting the best  $\beta$  and  $\sigma$  from a held-out validation set. We refrain from using the criterion  $\sigma = \varepsilon / \sqrt{d}$  for adversarial attacks, as they rely upon the extremal, as opposed to average, behavior of  $f$ . We check between  $\approx 10$  and  $\approx 30$  values of  $\sigma$  and  $\beta$  on the held-out validation set before selecting the values resulting in the most effective attacks. Attacks are performed using magnitudes  $\varepsilon \in \{0.25, 0.50, 0.75, 1.25, 1.75\}$  for MNIST and FMNIST and  $\varepsilon \in \{0.1, 0.2, 0.3, 0.4, 1.0\}$  for CIFAR10, which is easier to successfully attack at lower magnitudes due to its complexity. For more details see App. D.
Competing Methods: We compare SmoothHess with other gradient-based methods that model  $f$  locally; i.e. those which can be associated with the Taylor expansion around a smooth surrogate of  $f$  or  $f$  itself. Specifically, we compare with the Gradient and Hessian of SoftPlus smoothed network  $f_{\beta}$  [20, 39], SmoothGrad [76] and the vanilla (unsmoothed) Gradient. We also compare adversarial attacks with random vectors scaled to the attack magnitude  $\varepsilon$  as a baseline.
Figure 3: Evaluation of network predictions for two spirometry samples (Left and Right), using SmoothGrad and SmoothHess. Spirometry curves are plotted in blue. SmoothGrad (green) and SmoothHess (red) attributions are grouped into 0.5s intervals and plotted as bars for each interval. The plotted SmoothHess values are interactions with respect to the first 0.5s time interval. The red arrows point to the features that exhibit the largest interactions (endpoint) with the initial 0.5s (source). Sample 1 exhibits coughing within the first 4 seconds, as indicated by plateauing spirometry curve. We observe that the strongest interactions for Sample 1 occur at this initial cough-induced plateau. In contrast, Sample 2 exhibits no coughing within the first 4 seconds; the strongest interactions occur later near small fluctuations in the spirometry curve.
Additional experiments are provided in App. F. We compare the efficacy of SmoothHess with the unsmoothed Hessian (for adversarial attacks on SoftMax) and Swish [65] smoothed networks, both of which our method outperforms. We also run the  $\mathcal{P}_{MSE}$  experiment using a ResNet101 model, validating the superior ability of SmoothHess to capture interactions in larger networks.
Symmetric Smoothing - Four Quadrant Dataset: We investigate the ability of both SmoothHess and the SoftPlus Hessian to capture local feature interactions symmetrically. Results are shown in Figure 2. At each value of  $\sigma^2$ , aside from extremely small  $\sigma^2 < 1e-2.5$ , the off-diagonal element of SmoothHess is approximately equal to 2.5, the average interaction over the quadrants. The off-diagonal element of the SoftPlus Hessian is essentially never equal to 2.5. The results for SmoothHess follow from the fact that an isotropic covariance was used, the rotation invariance of which weights points that are equidistant from  $x$  equally. The SmoothHess off-diagonal element is not  $\approx 2.5$  at small  $\sigma$  because, despite the network being trained to memorize the data, such a small neighborhood will inherently reflect noise. The inability of SoftPlus to capture interactions symmetrically follows from the fact that smoothing is done internally on each neuron, which does not guarantee symmetry.
Perturbation Mean-Squared-Error: We show SmoothHess can be used to capture interactions occurring in a variety of neighborhoods around  $x$ . It is shown in Table 1 that SmoothHess + SmoothGrad (SH + SG) achieves the lowest  $\mathcal{P}_{MSE}$  for all 18 combinations of dataset, function and neighborhood size  $\varepsilon$ . We emphasize that this is despite the fact that only three values of  $\sigma$  were validated for SG + SH and SG on a held-out set compared to over one-hundred values of  $\beta$  for SP (H + G) and SP G. This indicates the superior ability of SmoothHess to model  $f$  over a diverse set of neighborhood sizes  $\varepsilon$ . Further, it can be seen that second-order methods SH + SG and SP (H + G) achieve significantly lower  $\mathcal{P}_{MSE}$  than their first-order counterparts, SG and SP G, respectively, sometimes by an order of magnitude. This confirms the intuition that higher order Taylor expansions of the smooth surrogate provide more accurate models of  $f$ . Interestingly, while SH + SG is clearly superior to SP (H + G), we see that SG and SP G are tied in many cases. This could indicate that the symmetric properties of Gaussian smoothing are comparatively more important when higher-order derivatives are considered. However, more investigation is needed.
Adversarial Attacks: We use the interactions found by SmoothHess to generate adversarial attacks on the classifier. We see from Table 2 that the Taylor expansion associated with SmoothHess and SmoothGrad generates the most powerful adversarial attacks for all datasets and attack magnitudes  $\varepsilon$ , aside from CIFAR10 at  $\varepsilon = 1.0$  where both methods successfully change most class predictions. The superiority of the SmoothHess attacks indicates that the Gaussian smoothing is capturing the extremal behavior of the predicted SoftMax probability more effectively than the other methods. One hypothesis for SmoothGrad outperforming SmoothHess in the largest neighborhood for CIFAR10 is that the network behavior is highly complex over a large area, and adding higher order terms decreases performance.
Qualitative Analysis of  $\mathbf{FEV}_1$  Prediction using Rejected Spirometry:
A spirometry test is a common procedure used to evaluate pulmonary function, and is the main diagnostic tool for lung diseases such as Chronic Obstructive Pulmonary Disease [40]. During an exam, the patient blows into a spirometer, which records the exhalation as a volume-time curve. Recent works have investigated the use of deep learning on spirograms for tasks such as subtyping [11], genetics [16, 91], and mortality prediction [36]. Additional background on spirometry is outlined in App. E.
Traditionally, exhalations interrupted by coughing are discarded for quality control reasons. We train a CNN on raw spirograms from the UK Biobank [79] to predict the patient's "Forced Expiratory Volume in 1 Second"  $\left(\mathrm{FEV}_{1}\right)$ , a metric frequently used to evaluate lung health, using efforts that were rejected due to coughing. In Fig. 3, we apply SmoothHess and SmoothGrad on two spirometry samples to understand their  $\mathrm{FEV}_{1}$  predictions. In Sample 1, coughing occurs within the first 4 seconds of the curve, as evidenced by the early plateauing of the curve which indicates a pause in exhalation. In contrast, Sample
Figure 4: Heatmap of the SmoothHess interactions for the spirometry sample in Fig. 3, calculated on CNNs with varying convolution kernel width.
2 exhibits no detectable coughing within the first 4 seconds. To improve the interpretability of the results, we group the features into 0.5 second time intervals.  $\mathrm{FEV}_1$  is traditionally measured in the initial 2 seconds of the non-rejected samples (see App. E), therefore we calculate SmoothHess interactions with respect to the first 0.5 second time interval.
$\mathrm{FEV}_1$  is known to be strongly affected by the presence of coughing [55]. The time intervals where coughing occurs, indicated by plateaus in the spirometry curves in Figure 3, should be an important signal for any model trained to predict  $\mathrm{FEV}_1$ . Indeed, we observe that for Sample 1, the SmoothGrad attribution for the first 0.5s interval is relatively low, with strong interactions occurring at the cough-induced plateaus. In contrast, the first 0.5s interval for Sample 2 shows high importance, with lower magnitude interactions that may be indicative of small fluctuations in the volume-time curve.
In Figure 4 we present the SmoothHess matrix for Sample 1 in Fig. 3, applied on two CNN models with different convolution kernel width. Interestingly, the smaller kernel width constrains the interactions to features that are spatially close together. In contrast, the features in the large kernel model have long-ranging interactions. These results agree with the intuition that smaller kernel widths, which constrain the receptive field of neurons up until the final layers, may correspondingly limit feature interactions.
We introduce SmoothHess, the Hessian of the network convolved with a Gaussian, as a method for quantifying second-order feature interactions. SmoothHess estimation, which relies only on gradient oracle calls, and cannot be performed with naive MC-averaging, is made possible by an extension of Stein's Lemma that we derive. We provide non-asymptotic complexity bounds for our estimation procedure. In contrast to previous works, our method can be run post-hoc, does not require architecture changes, affords the user localized flexibility over the weighting of input space, and can be run on any network output. We experimentally validate the superior ability of SmoothHess to capture interactions over a variety of neighborhoods compared to competing methods. Last, we use SmoothHess to glean insight into a network trained on real-world spirometry data.
Limitations: The outer product computations in SmoothHess estimation have space and time complexity  $\mathcal{O}(d^2)$ . When  $d \gg 0$  this becomes expensive: for instance ImageNet [19] typically has  $d^2 \approx 10^{10}$ . This is a common problem for all interaction effect methods. We leave as future work to explore computationally efficient alternatives or approximations to these outer products; a potential remedy is to devise an appropriate power method to estimate the top eigenvectors of SmoothHess, rather than its entirety, which fits well our estimation via sampling low-rank matrices.
This project was supported by NIH grants R01CA240771 and U24CA264369 from NCI, in part by MSKCC's Cancer Center core support NIH grant P30CA008748 from NCI, and NIH 2T32HL007427-41.