## Sohail BahmaniSchool of Electrical and Computer EngineeringGeorgia Institute of Technology [first name]DOT[last name]AT[ece.gatech.edu] |

## Research Vignette

Estimation of mean of a random variable is a fundamental problem of statistics since many estimation problems such as density estimation, regression, \(\ldots\), can be posed as mean estimation. The most benign and pedagogical example one can imagine is perhaps estimation of the mean of a random Gaussian vector in \(\mathbb{R}^d\), from a finite number of i.i.d. samples. The *sample mean*, the average of the observed samples, turns out to achieve the best possible *confidence interval* in the Gaussian setting. Unfortunately, the sample mean loses its glory as soon as we start dealing with heavy-tailed distributions or adversarially manipulated samples. In such unfavorable scenarios, the fundamental questions are then, “**How close can we get to the ideal confidence intervals?, With what kind of mean estimators?, and Whether such estimators are computationally tractable?**” There has been great strides in this line of research recently; we highly recommend the great survey papers [LM19] and [DK19] for more details on these recent advances as well as the historical background.

In [Bah20], we propose a competing mean estimator (in a separable Banach space) based on *empirical characteristic function*. For the sake of simpler exposition, we explain the ideas in the usual Euclidean setting. Given i.i.d. copies \(X_1,\dotsc,X_n\) of a random vector \(X\) in \(\mathbb{R}^d\), we want to estimate the mean of \(X\) denoted, by \(\mu_\star\) with error measured in the \(\ell_2\)-norm. **The only restriction on the law of \(X\) is that the corresponding covariance matrix, \(\varSigma_\star\), exists and is bounded. Furthermore, we may face strong contamination of the samples, where for some \(\eta\in(0,1/2)\), an adversary may arbitrarily manipulate up to \(\eta n\) samples.**

With the empirical characteristic function denoted by \[ \varphi_n(w) = \frac{1}{n}\sum_{i=1}^n \exp(\langle w, X_i\rangle)\nonumber\,, \] our proposed estimator is \[\widehat{\mu} \in \operatorname*{argmin}_\mu \max_{w\,:\,{\Vert w\Vert}\le r_n} \langle w, \mu\rangle - \mathrm{Im}(\varphi_n(w))\,,\nonumber\] for appropriately chosen radius \(r_n>0\). In particular, in the non-adversarial setting, we show that by choosing \(r_n = \frac{22\log(1/\delta)}{n\epsilon}\), with probability at least \(1-\delta\) the estimator achieves the bound \(\|\hat{\mu}-\mu_\star\|\le \epsilon\), if the prescribed \(\epsilon\) obeys \[\begin{align*} \epsilon & \ge \max\Big\{96\sqrt{\frac{\mathrm{tr}(\varSigma_\star)}{n}}+12\sqrt{\frac{\|\varSigma_\star\|_{\mathrm{op}}\log(1/\delta)}{n}},\\ & \hphantom{\max\Big\{ }\quad9{\left(\frac{\log(1/\delta)}{n}\right)}^{2/3}\|\mu_\star\|\Big\}\,.\end{align*}\] We also prove a similar statement in the adversarial setting. Basically, the effect of contamination in terms of the contamination parameter \(\eta\in(0,1/2)\) is an additional term of order \(\sqrt{\eta\|\varSigma_\star\|_{\mathrm{op}}}\) that appears in the confidence interval. Both of the derived confidence intervals have an undesirable dependence on \(\|\mu_\star\|\) that, even though vanishing at a rate \(n^{-2/3}\), prevents achieving the ideal purely sub-Gaussian rate. In [Bah20] we discuss ways to diminish this nuisance. Furthermore, we discuss how the estimator can be made oblivious to the accuracy level \(\epsilon\). The main drawback of the estimator is that, superficially, it does not appear to be computationally tractable.

#### Bibliography

[Bah20] S. Bahmani, “Nearly optimal robust mean estimation via empirical characteristic function,” arXiv preprint, 2020.

[LM19] G. Lugosi and S. Mendelson, “Mean estimation and regression under heavy-tailed distributions: A survey,” Foundations of Computational Mathematics, 19(5):1145–1190, 2019.

[DK8] I. Diakonikolas and D. Kane, “Recent advances in algorithmic high-dimensional robust statistics,” arXiv preprint, 2019.

*differentiable*and

*convex*function \(f\), and \(\boldsymbol{A}_\star\in\mathbb{R}^{n\times n}\), \(\boldsymbol{B}_\star\in\mathbb{R}^{n\times p}\) are model parametrs we want to estimate. This observation model generalizes some common

*recurrent neural net*(RNN) models. In particular, an RNN with

*Rectified Linear Unit*(ReLU) activations can be expressed in the above form by choosing \[ \begin{align*} f(\boldsymbol{x})\overset{\mathrm{\scriptscriptstyle def}}{=} \frac{1}{2}\sum_{i=1}^n {\left(\max\left\{x_i,0\right\}\right)}^2\,. \end{align*} \] Let \(\beta>0\) be a sufficiently large normalizing parameter. We can express the recursion \eqref{eq:RNN} equivalently in terms of \[ \begin{align*} \boldsymbol{C}_\star=\begin{bmatrix} \boldsymbol{A}_\star & \beta^{-1}\boldsymbol{B}_\star\end{bmatrix} \end{align*} \] and \[ \begin{align*} \boldsymbol{z}_t=\begin{bmatrix}\boldsymbol{x}_t\\ \beta\boldsymbol{u}_t\end{bmatrix}\,. \end{align*} \] Namely, for \(t\ge 1\) we have the recursion \[ \begin{align*} \boldsymbol{z}_t=\begin{bmatrix}\nabla f(\boldsymbol{C}_\star\boldsymbol{z}_{t-1})\\ \beta \boldsymbol{u}_t \end{bmatrix}\,, \end{align*} \] and at \(t=0\) we have \[ \begin{align*} \boldsymbol{z}_0=\begin{bmatrix} \boldsymbol{0}\\ \beta \boldsymbol{u}_0 \end{bmatrix}\,. \end{align*} \] It follows from the convexity of \(f\) and \eqref{eq:RNN} that \[ \begin{align*} f(\boldsymbol{C}\boldsymbol{z}_{t-1})-\langle\boldsymbol{x}_t, \boldsymbol{C}\boldsymbol{z}_{t-1}\rangle \ge f(\boldsymbol{C}_\star\boldsymbol{z}_{t-1})-\langle\boldsymbol{x}_t, \boldsymbol{C}_\star\boldsymbol{z}_{t-1} \rangle\,. \end{align*} \] Therefore, we can formulate a convex program as an estimator for \(\boldsymbol{C}_\star\): \[ \begin{align*} \widehat{\boldsymbol{C}} = \operatorname*{argmin}_{\boldsymbol{C}}\ \sum_{t=1}^T f(\boldsymbol{C}\boldsymbol{z}_{t-1})-\langle\boldsymbol{x}_t, \boldsymbol{C}\boldsymbol{z}_{t-1}\rangle\,. \end{align*} \] In [BR19], under some additional assumptions on the nonlinearity and the distribution of the input, we showed that, with high probability, a time horizon \(T\) that scales with \(n+p\), up to logarithmic factors, suffices for exact recovery of \(\boldsymbol{C}_\star\) from a single trajectory.

#### Bibliography

[BR19] S. Bahmani and J. Romberg, “Convex programming for estimation in nonlinear recurrent models,” 2019.

**difference of convex**(DC) observation functions. This class of regression problems are much broader than those with the ``convex observation'' model considered in Part (I) because many non-convex functions can be written in DC form as well. In the considered observation model, we measure \(\boldsymbol{x}_\star\in\mathbb{R}^N\) indirectly as \begin{align*} y_m & = f_m(\boldsymbol{x}_\star) - g_m(\boldsymbol{x}_\star) + \xi_m\,, & m=1,\dotsc,M\,, \end{align*} where the pairs of functions \((f_m,\,g_m)\) are i.i.d. copies of a pair of random

*convex*functions \((f,\,g)\), and \(\xi_m\) denotes the noise as in Part (I). Assuming that we have access to \begin{align} \boldsymbol{a}_0 & \approx \frac{1}{2M}\sum_{m=1}^M\nabla f_m(\boldsymbol{x}_\star) + \nabla g_m(\boldsymbol{x}_\star)\,,\label{a0-approx} \end{align} where the approximation has a certain precise form, then we formulate the estimator \(\widehat{\boldsymbol{x}}\) of \(\boldsymbol{x}_\star\) as \begin{align*} \widehat{\boldsymbol{x}} & = \operatorname*{argmax}_{\boldsymbol{x}}\ \langle\boldsymbol{a}_0, \boldsymbol{x}\rangle - \frac{1}{M}\sum_{m=1}^M \max \left\lbrace f_m(\boldsymbol{x}) - y_m, g_m(\boldsymbol{x}) \right\rbrace\,. \end{align*} Under mild

*non-degeneracy*and

*regularity*assumptions, we established in [Bah18] a sample complexity for guaranteed accuracy of \(\widehat{\boldsymbol{x}}\) using a PAC-Bayesian analysis. To provide a concrete example we also studied

*bilinear regression*with standard Gaussian factors as a special case; for this problem we also described a method to create a vector \(\boldsymbol{a}_0\) that provably satisfies the required approximation \eqref{a0-approx}.

#### Bibliography

[Bah18] S. Bahmani, “Estimation from nonlinear observations via
convex programming, with application to bilinear regression,” *Elect. J.
Statistics*, 13(1): 1978–2011, 2019.

#### Problem statement

Consider the problem of estimating an \(N\)-dimensional signal \(\boldsymbol{x}_\star\) from
observations of the form
\begin{align}
y_m & = f_m(\boldsymbol{x}_\star) + \xi_m\,, & m=1,2,\dotsc,M\,,
\label{cvxreg}
\end{align}
where the functions \(f_m\) are i.i.d. copies of a randomly drawn convex function \(f\), and the noise terms
are represented by \(\xi_m\). Despite the convexity assumption on the functions \(f_m\), the observation
model \eqref{cvxreg} is quite general and includes many standard statistical models as the special case
including *generalized linear models* and *single hidden layer neural nets*. I addressed the
general regression problem \eqref{cvxreg} in [BR17b] building upon ideas I developed in
[BR17a] to address a special case often known as *phase retrieval*. It is more
illustrative to begin my explanation with this special case as well.

#### A new approach to phase retrieval

The problem of phase retrieval appears in areas such as imaging and optics where the sensors often measure only intensities; the sign- (or phase-) information is generally lost. The observation model in the phase retrieval problem, assuming no noise in the measurements, can be abstracted as the system of quadratic equations \begin{equation*} \begin{aligned} y_1 & = |\boldsymbol{a}_1^*\boldsymbol{x}_\star|^2\\ y_2 & = |\boldsymbol{a}_2^*\boldsymbol{x}_\star|^2\\ \vdots & \qquad\vdots\\ y_M & = |\boldsymbol{a}_M^*\boldsymbol{x}_\star|^2\,, \end{aligned} \end{equation*} for \(\boldsymbol{x}_\star\in\mathbb{C}^N\). Clearly, this model corresponds to \eqref{cvxreg} with \(f_m(\boldsymbol{x}) = |\boldsymbol{a}_m^*\boldsymbol{x}|^2\) for i.i.d. draws of \(\boldsymbol{a}_m\), and \(\xi_m=0\). In [BR17a], I formulated a new estimator for this problem as the convex program \begin{equation} \begin{aligned} \operatorname*{argmax}_{\boldsymbol{x}}\ & \mathrm{Re}(\boldsymbol{a}_0^*\boldsymbol{x}) & \\ \text{subject to}\ & |\boldsymbol{a}_m^*\boldsymbol{x}|^2 \le y_m, & m=1,2,\dotsc,M\,, \end{aligned} \label{linmax} \end{equation} where \(\boldsymbol{a}_0\) denotes an “anchor vector” that obeys \begin{equation*} |\boldsymbol{a}_0^*\boldsymbol{x}_\star| \ge \delta \left\lVert\boldsymbol{a}_0\right\rVert_2 \left\lVert\boldsymbol{x}_\star\right\rVert_2\,, \end{equation*} for some absolute constant \(\delta \in (0,1]\). The anchor vector can be constructed from random observations similar to initializations in some non-convex methods (e.g., Wirtinger Flow); the details can be found in the paper. Here, I explain the geometric intuition behind \eqref{linmax} in the case of real-valued variables for clarity. As illustrated in Figure 1, each of the constraints in \eqref{linmax} form a slab of feasible points, whose intersection is a convex polytope \(\mathcal{K}\). Clearly, \(\boldsymbol{x}_\star\) is an extreme point of \(\mathcal{K}\). The solution to \eqref{linmax} is also always an extreme point of \(\mathcal{K}\), since it is a solution to linear maximization over the convex body \(\mathcal{K}\). The key observation is that if the anchor vector \(\boldsymbol{a}_0\) has a non-trivial component in the direction of \(\boldsymbol{x}_\star\), we can expect that the extreme point found by \eqref{linmax} coincides with \(\boldsymbol{x}_\star\).

Using classic results from *statistical learning theory*, I showed that with high probability
\begin{equation*}
M = C_\delta N
\end{equation*}
independent random measurements suffice to recover \(\boldsymbol{x}_\star\)
using \eqref{linmax}, with \(C_\delta\) being an absolute constant depending only on \(\delta\). Robustness
under specific noise models is also addressed in the paper.

#### Why does \eqref{linmax} matter?

Previous convex relaxations for phase retrieval (e.g., PhaseLift, and PhaseCut) were based on the idea of *lifting* and
*semidefinite programming* (SDP). While lifting-based methods are technically computationally
tractable, their dependence on SDP prohibits their scalability. In contrast, \eqref{linmax} operates in the
natural domain of the problem and competes with non-convex methods for phase retrieval (e.g., Wirtinger Flow). It also benefits from versatility,
flexibility, and robustness that is associated with convex programming. More importantly, as discussed
below, the principles used in formulation and analysis of \eqref{linmax} apply in a more general setting.

#### What about the general case \eqref{cvxreg}?

In [BR17b], I proposed the convex program \begin{equation} \begin{aligned} \operatorname*{argmax}_{\boldsymbol{x}}\ & \langle \boldsymbol{a}_0,\boldsymbol{x}\rangle \\ \text{subject to}\ & \sum_{m=1}^M \max\{f_m(\boldsymbol{x})-y_m, 0\} \le \varepsilon\, , \end{aligned} \label{anchored_reg} \end{equation} where \(\boldsymbol{a}_0\) is an “anchor vector” that obeys \begin{equation*} \langle\boldsymbol{a}_0,\boldsymbol{x}_\star\rangle \ge \delta \left\lVert\boldsymbol{a}_0\right\rVert_2\left\lVert\boldsymbol{x}_\star\right\rVert_2\,, \end{equation*} as an estimator for the general regression problem \eqref{cvxreg}. Some schemes for constructing the anchor from the measurements are described in the paper, but we omit the discussion for brevity. Furthermore, to avoid technical details, here I state the main result of the paper (i.e., Theorem 2.1) on the sample complexity of \eqref{anchored_reg} in an informal way. In the proved bound, there are two important quantities. The first quantity, \(\mathfrak{C}_M(\mathcal{A}_\delta)\), measures the “size” of a set \(\mathcal{A}_\delta\) that depends only on \(\boldsymbol{x}_\star\) and \(\delta\), with respect to the randomness in the gradients \(\nabla f_m(\boldsymbol{x}_\star)\). The second quantity, \(p_\tau(\mathcal{A}_\delta)\), is some measure of the “eccentricity” of the random vector \(\nabla f_m(\boldsymbol{x}_\star)\) with respect to the set \(\mathcal{A}_\delta\) in terms of a parameter \(\tau\). Ignoring some details, the result established in [BR17b] basically states that \begin{equation*} M \gtrsim \left(\frac{\mathfrak{C}_M(\mathcal{A}_\delta)}{\tau p_\tau(\mathcal{A}_\delta)}\right)^2\,, \end{equation*} measurements are sufficient to guarantee that \eqref{anchored_reg} yields an accurate estimate.

#### Bibliography

[BR17a] S. Bahmani and J. Romberg, “Phase
retrieval meets statistical learning theory: A flexible
convex relaxation,” *In Proceedings of the
20th International Conference on Artificial Intelligence
and Statistics (AISTATS'17)*, vol. 54 of Proceedings
of Machine Learning Research , pp. 252–260.

[BR17b] S. Bahmani and J. Romberg, “Solving equations of random convex functions
via anchored regression,” *Foundations of Computational Mathematics*,
19(4):813–841, 2019.

An important characteristic of networks in many applications is their connectivity which is often a
crucial factor in the performance of the network. An interesting and important problem is then to measure
robustness of the connectivity under some form of perturbation of the network. *Site percolation*, or
simply random removal of nodes as illustrated in Figure 2, is one of these
perturbation models that is studied in mathematics and statistical physics.

*Algebraic connectivity* of a graph is a an analytical measure of connectivity that is also related to
the conductance of the graph through the Cheeger's inequality. Formally, the algebraic connectivity of a
graph with the adjacency matrix \(\boldsymbol{A}\) can be defined as *the second smallest eigenvalue*
of the graph Laplacian \(\boldsymbol{L} = \boldsymbol{D} - \boldsymbol{A}\) where
\(\boldsymbol{D}=\mathrm{diag}(\boldsymbol{A}\boldsymbol{1})\) is the diagonal matrix of the vertex degrees.
In [BRT18], using tools from random matrix theory I derived a lower bound for algebraic
connectivity of a graph that survives from a generally non-homogeneous site percolation. In the special case
of homogeneous site percolation over a certain class of regular graphs, our analytical result virtually
coincides with the state-of-the-art that is established using refined combinatorial arguments.

#### Bibliography

[BRT18] S. Bahmani, J. Romberg, and P. Tetali, “Algebraic
connectivity under site percolation in finite weighted graphs,” *IEEE Trans. Network
Science and Engineering*, 5(2):86–91, 2018.

## Publications

- In Review/Revision
- Journal Paper
- Conference Paper
- Technical Report

#### In Review/Revision

- S. Kim, S. Bahmani, K. Lee, “Max-Linear Regression by Scalable and Guaranteed Convex Programming,” in review, 2021.

arXiv

#### 2021

- S. Bahmani, K. Lee, “Low-rank matrix estimation from rank-one projections by unlifted convex optimization”, to appear in the
*SIAM J. on Matrix Analysis and Applications*, 2021.

arXiv - S. Bahmani, “Nearly optimal robust mean estimation via empirical characteristic function,”
*Bernoulli*, 27(3): 2139–2158, 2021.

arXivProj. Euclid

#### 2020

- S. Bahmani, J. Romberg, “Convex programming for estimation in
nonlinear recurrent models,”
*Journal of Machine Learning Research*, (235):1–20, 2020.

arXivJMLRCode - K. Lee, S. Bahmani, J. Romberg, Y. Eldar, “Phase retrieval of
low-rank matrices by anchored regression,”
*Information and Inference: A Journal of the IMA*, 2020.

arXivOxford Journals

#### 2019

- S. Bahmani, “Estimation from nonlinear observations via convex
programming, with application to bilinear regression,”
*Electronic J. of Statistics*, 13(1): 1978–2011. 2019.

arXivProj. Euclid

#### 2018

- S. Bahmani and J. Romberg, “Solving
equations of random convex functions via anchored regression,”
*Foundations of Computational Mathematics*, 19(4):813–841, 2019.

arXivSpringer - S. Bahmani, J. Romberg, P. Tetali, “Algebraic
connectivity under site percolation in finite weighted graphs,”
*IEEE Trans. on Network Science and Engineering*, 5(2):86–91, 2018. arXivIEEEXplore

#### 2017

- S. Bahmani and J. Romberg, “A
flexible convex relaxation for phase retrieval,”
*Electronic Journal of Statistics,*11(2):5254–5281, 2017. (This is an extended version of the AISTATS’17 paper.)

Proj. Euclid - S. Bahmani and J. Romberg, “Phase
retrieval meets statistical learning theory: A flexible convex
relaxation,”
*In Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS'17)*, vol. 54 of Proceedings of Machine Learning Research , pp. 252–260. (**Best paper award**) arXivPMLR

#### 2016

- S. Bahmani and J. Romberg, “Near-optimal
estimation of simultaneously sparse and low-rank matrices from
nested linear measurements,”
*Information and Inference: A Journal of the IMA*5(3):331–351, 2016.

arXivOxford Journals - S. Bahmani, P. Boufounos, and B. Raj, “Learning
model-based sparsity via projected gradient descent,”
*IEEE Trans. Info. Theory*, 62(4):2092–2099, 2016.

arXivIEEEXplore

#### 2015

- S. Bahmani and J. Romberg, “Sketching
for simultaneously sparse and low-rank covariance matrices,”
in
*Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP'15), IEEE 6th International Workshop on*, pp. 357–360, Cancun, Mexico, Dec. 2015.arXivIEEEXplore - S. Bahmani and J. Romberg,
“Efficient compressive phase retrieval with constrained
sensing vectors,” in
*Advances in Neural Information Processing Systems (NIPS'15),*vol. 28, pp. 523–531, Montréal, Canada, Dec. 2015.

arXivNIPS - S. Bahmani and J. Romberg, “Lifting
for blind deconvolution in random mask imaging:
Identifiability and convex relaxation,”
*SIAM Journal on Imaging Sciences*, 8(4):2203–2238, 2015.

arXivSIAM - S. Bahmani and J. Romberg, “Compressive
deconvolution in random mask imaging,”
*IEEE Trans. on Computational Imaging*, 1(4):236–246, 2015.

arXivIEEEXplore

#### 2013

- S. Bahmani, B. Raj, and P. T. Boufounos, “Greedy
sparsity-constrained optimization,”
*Journal of Machine Learning Research*, 14(3):807–841, 2013.

arXivJMLRCode - S. Bahmani, P. Boufounos, and B. Raj, “Robust
1-bit compressive sensing via gradient support pursuit,”
Apr. 2013.

arXiv

#### 2012

- S. Bahmani, B. Raj, “A
unifying analysis of projected gradient descent for
\(\ell_p\)-constrained least squares,”
*Applied and Computational Harmonic Analysis*, 34(3):366–378, 2012.

arXivElsevier

#### 2011

- S. Bahmani, P. Boufonos, and B. Raj, “Greedy
sparsity-constrained optimization,” in
*Conf. Record of the 45th Asilomar Conference on Signals, Systems, and Computers (ASILOMAR'11)*, pp. 1148–1152, Pacific Grove, CA, Nov. 2011. IEEEXploreSlidesCode

#### 2010

- S. Bahmani, I. Bajić, and A. HajShirmohammadi, “Joint decoding of
unequally protected
JPEG2000 images and Reed-Solomon codes,”
*IEEE Trans. Image Processing*, 19(10):2693–2704, Oct. 2010.

IEEEXplore

#### 2009

- S. Bahmani, I. Bajić, and A. HajShirmohammadi, “Improved joint
source channel decoding of
JPEG2000 images and Reed-Solomon codes,”
*Proc. IEEE ICC'09*, Dresden, Germany, Jun. 2009.

IEEEXplore

#### 2008

- S. Bahmani, I. Bajić, A. HajShirmohammadi, “Joint
source channel decoding of JPEG2000 images with unequal loss
protection,”
*Proc. IEEE ICASSP'08*, pp. 1365–1368, Las Vegas, NV, Mar. 2008.

IEEEXplore

#### Thesis

- S. Bahmani, Algorithms
for sparsity-constrained optimization, PhD dissertation,
Department of Electrical & Computer Engineernig, Carnegie
Mellon University, Pittsburgh, PA, Feb. 2013.

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