Maximum Likelihood Matrix Completion Under Sparse Factor Models: - - PowerPoint PPT Presentation

Maximum Likelihood Matrix Completion Under Sparse Factor Models: Error Guarantees and Efficient Algorithms

Jarvis Haupt

Department of Electrical and Computer Engineering University of Minnesota

Institute for Computational and Experimental Research in Mathematics (ICERM) Workshop on Approximation, Integration, and Optimization October 1, 2014

Background and Motivation Problem Statement Error Bounds Algorithmic Approach Experimental Results Acknowledgments

Section 1 Background and Motivation

Background and Motivation Problem Statement Error Bounds Algorithmic Approach Experimental Results Acknowledgments

A Classical Example

Sampling Theorem:

(Whittaker/Kotelnikov/Nyquist/Shannon, 1930’s-1950’s)

Original Signal (Red) Samples (Black) Accurate Recovery (and Imputation) via Ideal Low-Pass Filtering when Original Signal is Bandlimited Basic “Formula” for Inference: To draw inferences from limited data (or here, to impute missing elements), need to leverage underlying structure in the signal being inferred.

Background and Motivation Problem Statement Error Bounds Algorithmic Approach Experimental Results Acknowledgments

A Contemporary Example

Matrix Completion:

(Candes & Recht; Keshavan, et al.; Candes & Tao; Candes & Plan; Negahban & Wainwright; Koltchinskii et al.; Davenport et al.;... 2009- )

Samples Accurate Recovery (and Imputation) via Convex Optimization when Original Matrix is Low-Rank Low-rank modeling assumption commonly utilized in collaborative filtering applications (e.g. the Netflix prize), to describe settings where each

• bserved value depends on only a

few latent factors or features.

Background and Motivation Problem Statement Error Bounds Algorithmic Approach Experimental Results Acknowledgments

Beyond Low Rank Models?

Low-Rank Models: All columns of the ma- trix are well-approximated as vectors in common linear subspace. Union of Subspaces Model: All columns of the matrix are well-approximated as vectors in a union of linear subspaces. Union of subspaces models are at the essence of sparse subspace clustering (Elhamifar & Vidal;

Soltanolkotabi et al.; Erikkson et al; Balzano et al) and dictionary learning (Olshausen & Field; Aharon et al; Mairal et al.;...).

Here, we examine the efficacy of such models in matrix completion tasks.

Background and Motivation Problem Statement Error Bounds Algorithmic Approach Experimental Results Acknowledgments

Section 2 Problem Statement

Background and Motivation Problem Statement Error Bounds Algorithmic Approach Experimental Results Acknowledgments

“Sparse Factor” Data Models

We assume the unknown X∗ ∈ Rn1×n2 we seek to estimate admits a factorization of the form X∗ = D∗A∗, D∗ ∈ Rn1×r, A∗ ∈ Rr×n2 where

• D∗max maxi,j |Di,j| ≤ 1

(essentially to fix scaling ambiguities)

• A∗max ≤ Amax for a constant 0 < Amax ≤ (n1 ∨ n2)
• X∗max ≤ Xmax/2 for a constant Xmax ≥ 1

Background and Motivation Problem Statement Error Bounds Algorithmic Approach Experimental Results Acknowledgments

“Sparse Factor” Data Models

We assume the unknown X∗ ∈ Rn1×n2 we seek to estimate admits a factorization of the form X∗ = D∗A∗, D∗ ∈ Rn1×r, A∗ ∈ Rr×n2 where

• D∗max maxi,j |Di,j| ≤ 1

(essentially to fix scaling ambiguities)

• A∗max ≤ Amax for a constant 0 < Amax ≤ (n1 ∨ n2)
• X∗max ≤ Xmax/2 for a constant Xmax ≥ 1

Our Focus: Sparse factor models, characterized by (approximately or exactly) sparse A∗.

Background and Motivation Problem Statement Error Bounds Algorithmic Approach Experimental Results Acknowledgments

Observation Model

We observe X∗ only at a subset S ∈ {1, 2, . . . , n1} × {1, 2, . . . , n2} of its locations. For some γ ∈ (0, 1] each (i, j) is in S independently with probability γ, and interpret γ = m(n1n2)−1, so that m = is the nominal number of observations. Observations {Yi,j}(i,j)∈S YS conditionally independent given S, modeled via joint density pX∗

S (YS) =

• (i,j)∈S

pX ∗

i,j (Yi,j)

• scalar densities

Background and Motivation Problem Statement Error Bounds Algorithmic Approach Experimental Results Acknowledgments

Estimation Approach

We estimate X∗ via a sparsity-penalized maximum likelihood approach: for λ > 0, we take

• X = arg

min

X=DA∈X

• − log pXS (YS) + λ · A0
• .

Background and Motivation Problem Statement Error Bounds Algorithmic Approach Experimental Results Acknowledgments

Estimation Approach

We estimate X∗ via a sparsity-penalized maximum likelihood approach: for λ > 0, we take

• X = arg

min

X=DA∈X

• − log pXS (YS) + λ · A0
• .

The set X of candidate reconstructions is any subset of X ′, where X ′ {X = DA : D ∈ D, A ∈ A, Xmax ≤ Xmax} , where

• D: the set of all matrices D ∈ Rn1×r whose elements are discretized to one of L

uniformly-spaced values in the range [−1, 1]

• A: the set of all matrices A ∈ Rr×n2 whose elements either take the value zero, or are

discretized to one of L uniformly-spaced values in the range [−Amax, Amax]

Background and Motivation Problem Statement Error Bounds Algorithmic Approach Experimental Results Acknowledgments

Section 3 Error Bounds

Background and Motivation Problem Statement Error Bounds Algorithmic Approach Experimental Results Acknowledgments

A General “Sparse Factor” Matrix Completion Error Guarantee

Theorem (A. Soni, S. Jain, J.H., and S. Gonella, 2014) Let β > 0 and set L = (n1 ∨ n2)β. If CD satisfies CD ≥ maxX∈X maxi,j D(pX ∗

i,j pXi,j ), then for

any λ ≥ 2 · (β + 2) ·

• 1 + 2CD

3

• · log(n1 ∨ n2), the sparsity penalized ML estimate
• X = arg

min

X=DA∈X

• − log pXS (YS) + λ · A0
• satisfies the (normalized, per-element) error bound

ES,YS

• −2 log A(p

X, pX∗)

• n1n2

≤ 8CD log m m +3 min

X=DA∈X

D(pX∗pX) n1n2 +

• λ + 4CD(β + 2) log(n1 ∨ n2)

3 n1p + A0 m

• .

Here:

A(pX, pX∗)

i,j A(pXi,j , pX∗ i,j ) where A(pXi,j , pX∗ i,j ) EpX∗ i,j

• pXi,j /pX∗

i,j

• is the Hellinger Affinity

D(pX∗pX)

i,j D(pX∗ i,j pXi,j ) where D(pX∗ i,j pXi,j ) EpX∗ i,j

• log(pX∗

i,j /pXi,j )

• is KL Divergence

Background and Motivation Problem Statement Error Bounds Algorithmic Approach Experimental Results Acknowledgments

A General “Sparse Factor” Matrix Completion Error Guarantee

Theorem (A. Soni, S. Jain, J.H., and S. Gonella, 2014) Let β > 0 and set L = (n1 ∨ n2)β. If CD satisfies CD ≥ maxX∈X maxi,j D(pX ∗

i,j pXi,j ), then for

any λ ≥ 2 · (β + 2) ·

• 1 + 2CD

3

• · log(n1 ∨ n2), the sparsity penalized ML estimate
• X = arg

min

X=DA∈X

• − log pXS (YS) + λ · A0
• satisfies the (normalized, per-element) error bound

ES,YS

• −2 log A(p

X, pX∗)

• n1n2

≤ 8CD log m m +3 min

X=DA∈X

D(pX∗pX) n1n2 +

• λ + 4CD(β + 2) log(n1 ∨ n2)

3 n1p + A0 m

• .

Here:

A(pX, pX∗)

i,j A(pXi,j , pX∗ i,j ) where A(pXi,j , pX∗ i,j ) EpX∗ i,j

• pXi,j /pX∗

i,j

• is the Hellinger Affinity

D(pX∗pX)

i,j D(pX∗ i,j pXi,j ) where D(pX∗ i,j pXi,j ) EpX∗ i,j

• log(pX∗

i,j /pXi,j )

• is KL Divergence

Next, we instantiate this result for some specific cases (using a specific choice of β, λ).

Background and Motivation Problem Statement Error Bounds Algorithmic Approach Experimental Results Acknowledgments

Additive White Gaussian Noise Model

Suppose each observation is corrupted by zero-mean AWGN with known variance σ2, so that pX∗

S (YS) =

1 (2πσ2)|S|/2 exp  − 1 2σ2

• (i,j)∈S

(Yi,j − X ∗

i,j)2

  . Let X = X ′, essentially (a discretization of) a set of rank and max-norm constrained matrices. Gaussian Noise (Exact Sparse Factor Model) If A∗ is exactly sparse with A∗0 nonzero elements, the sparsity penalized ML estimate satisfies ES,YS

• X∗ −

X2

F

• n1n2

= O

• (σ2 + X2

max)

n1r + A∗0 m

• log(n1 ∨ n2)
• .

Background and Motivation Problem Statement Error Bounds Algorithmic Approach Experimental Results Acknowledgments

AWGN – Our Result in Context

Gaussian Noise (Exact Sparse Factor Model) If A∗ is exactly sparse with A∗0 nonzero elements, the sparsity penalized ML estimate satisfies ES,YS

• X∗ −

X2

F

• n1n2

= O

• (σ2 + X2

max)

n1r + A∗0 m

• log(n1 ∨ n2)
• .

Compare with result of (Koltchinskii et al, 2011); when X∗ is max-norm and rank-constrained, nuclear-norm penalized optimization yields estimate satisfying X∗ − X2

F

n1n2 = O

• (σ2 + X2

max)

(n1 + n2)r m

• log(n1 ∨ n2)
• with high probability.

Note: Our guarantees can have improved error performance in the case where A∗0 ≪ n2r. The two bounds coincide when A∗ is not sparse (take A∗0 = n2r in our error bounds).

Background and Motivation Problem Statement Error Bounds Algorithmic Approach Experimental Results Acknowledgments

AWGN Model (Extension to Approximately Sparse Factor Model)

Recall: For p ≤ 1, a vector x ∈ Rn is said to belong to a weak-ℓp ball of radius R > 0, denoted x ∈ wℓp(R), if its ordered elements |x(1)| ≥ |x(2)| ≥ · · · ≥ |x(n)| satisfy |x(i)| ≤ Ri−1/p for all i ∈ {1, 2, . . . , n}. With this, we can state a variant of the above for when columns of A∗ are approximately sparse.

Background and Motivation Problem Statement Error Bounds Algorithmic Approach Experimental Results Acknowledgments

AWGN Model (Extension to Approximately Sparse Factor Model)

Recall: For p ≤ 1, a vector x ∈ Rn is said to belong to a weak-ℓp ball of radius R > 0, denoted x ∈ wℓp(R), if its ordered elements |x(1)| ≥ |x(2)| ≥ · · · ≥ |x(n)| satisfy |x(i)| ≤ Ri−1/p for all i ∈ {1, 2, . . . , n}. With this, we can state a variant of the above for when columns of A∗ are approximately sparse. Gaussian Noise (Approximately Sparse Factor Model) Consider the same Gaussian noise model described above. If for some p ≤ 1 all columns of A∗ belong to a weak-ℓp ball of radius Amax, then for α = 1/p − 1/2, ES,YS

• X∗ −

X2

F

• n1n2

= O

• A2

max

n2 m

• 2α

2α+1 + (σ2 + X2

max)

n1r m + n2 m

• 2α

2α+1

• log(n1 ∨ n2)
• Note:

n2

m

• 2α

2α+1 ≤ n2m− 2α 2α+1 ⇐ aggregate error of estimating n2 vectors in wℓp from noisy obs.

Background and Motivation Problem Statement Error Bounds Algorithmic Approach Experimental Results Acknowledgments

Additive Laplace Noise Model

Suppose each observation is corrupted by additive Laplace noise with known parameter τ > 0, so pX∗

S (YS) =

τ 2 |S| exp  −τ

• (i,j)∈S

|Yi,j − X ∗

i,j|

  . Let X = X ′, essentially (a discretization of) a set of rank and max-norm constrained matrices. Laplace Noise (Exact Sparse Factor Model) If A∗ is exactly sparse with A∗0 nonzero elements, the sparsity penalized ML estimate satisfies ES,YS

• X∗ −

X2

F

• n1n2

= O

• 1

τ 2 + X2

max

• O(variance + X2

max)

τXmax n1r + A∗0 m

• “parametric-like” form

similar to sparse model Gaussian-noise case log(n1∨n2)

• .

Can also obtain results for the approximately sparse case here, analogously to above...

Background and Motivation Problem Statement Error Bounds Algorithmic Approach Experimental Results Acknowledgments

Poisson-distributed Observations

Suppose that each element of X∗ satisfies X ∗

i,j ≥ Xmin for some Xmin > 0, and that each

• bservation is Poisson-distributed, so that YS ∈ N|S| and

pX∗

S (YS) =

• (i,j)∈S

(X ∗

i,j)Yi,j e−X ∗

i,j

(Yi,j)! , Let X = {X ∈ X ′ : Xi,j ≥ 0 for all (i, j) ∈ {1, 2, . . . , n1} × {1, 2, . . . , n2}}. (To allow only non-negative rate estimates) Poisson-distributed Observations (Exact Sparse Factor Model) If A∗ is exactly sparse with A∗0 nonzero elements, the sparsity penalized ML estimate satisfies ES,YS

• X∗ −

X2

F

• n1n2

= O

• Xmax + X2

max

Xmax Xmin

• O(worst-case variance + X2

max)

when Xmax/Xmin = O(1) n1r + A∗0 m

• log(n1 ∨n2)
• .

Can also obtain results for the approximately sparse case here, analogously to above...

Background and Motivation Problem Statement Error Bounds Algorithmic Approach Experimental Results Acknowledgments

One-bit Observations

Let link function F : R → [0, 1] be a differentiable link function with f (t) = d

dt F(t). Suppose

each observation Yi,j for (i, j) ∈ S is Bernoulli(F(X ∗

i,j))-distributed, so that

pX∗

S (YS) =

• (i,j)∈S
• F(X ∗

i,j)

Yi,j 1 − F(X ∗

i,j)

1−Yi,j Assume F(Xmax) < 1, F(−Xmax) > 0, and inf|t|≤Xmax f (t) > 0. One-bit Observations (Exact Sparse Factor Model) If A∗ is exactly sparse with A∗0 nonzero elements, the sparsity penalized ML estimate satisfies ES,YS

• X∗ −

X2

F

• n1n2

= O

• cF,Xmax

c′

F,Xmax

1 cF,Xmax + X2

max

n1r + A∗0 m

• log(n1 ∨ n2)
• ,

where cF,Xmax

• sup

|t|≤Xmax

1 F(t)(1 − F(t))

• ·
• sup

|t|≤Xmax

f 2(t)

• c′

F,Xmax

• inf

|t|≤Xmax

f 2(t) F(t)(1 − F(t)) . Can also obtain results for the approximately sparse case here, analogously to above...

Background and Motivation Problem Statement Error Bounds Algorithmic Approach Experimental Results Acknowledgments

Comparisons to “One bit Matrix Completion”

One-bit Observations (Exact Sparse Factor Model) If A∗ is exactly sparse with A∗0 nonzero elements, the sparsity penalized ML estimate satisfies ES,YS

• X∗ −

X2

F

• n1n2

= O

• cF,Xmax

c′

F,Xmax

1 cF,Xmax + X2

max

n1r + A∗0 m

• log(n1 ∨ n2)
• ,

Compare with low-rank recovery result of (Davenport et al., 2012); maximum likelihood

• ptimization over a set of max-norm and nuclear-norm constrained candidates yields estimate

satisfying X∗ − X2

F

n1n2 = O

• CF,XmaxXmax
• (n1 + n2)r

m

• with high probability, where CF,Xmax analogous to (cF,Xmax/c′

F,Xmax) factor in our bounds.

Extra loss of Xmax log(n1 ∨ n2) in our bound, but faster “parametric-like” dependence on m (in addition to the “sparse factor” improvement). Lower bounds for “sparse factor” model still

• pen (we think!).

Background and Motivation Problem Statement Error Bounds Algorithmic Approach Experimental Results Acknowledgments

Section 4 Algorithmic Approach

Background and Motivation Problem Statement Error Bounds Algorithmic Approach Experimental Results Acknowledgments

A Non-Convex Problem...

Our optimizations take the general form min

D∈Rn1×r ,A∈Rr×n2 ,X∈Rn1×n2

• i,j

−si,j logpXi,j (Yi,j) + IX (X) + ID(D) + IA(A) + λA0 s.t. X = DA. where si,j = 1 if (i, j) ∈ S (and 0 otherwise) and IX (.), ID(.), IA(.) are indicator functions.

Background and Motivation Problem Statement Error Bounds Algorithmic Approach Experimental Results Acknowledgments

A Non-Convex Problem...

Our optimizations take the general form min

D∈Rn1×r ,A∈Rr×n2 ,X∈Rn1×n2

• i,j

−si,j logpXi,j (Yi,j) + IX (X) + ID(D) + IA(A) + λA0 s.t. X = DA. where si,j = 1 if (i, j) ∈ S (and 0 otherwise) and IX (.), ID(.), IA(.) are indicator functions. Multiple sources of non-convexity:

• ℓ0 regularizer
• discretized sets D and A
• inherent bilinearity of the model!

Background and Motivation Problem Statement Error Bounds Algorithmic Approach Experimental Results Acknowledgments

A Non-Convex Problem...

Our optimizations take the general form min

D∈Rn1×r ,A∈Rr×n2 ,X∈Rn1×n2

• i,j

−si,j logpXi,j (Yi,j) + IX (X) + ID(D) + IA(A) + λA0 s.t. X = DA. where si,j = 1 if (i, j) ∈ S (and 0 otherwise) and IX (.), ID(.), IA(.) are indicator functions. Multiple sources of non-convexity:

• ℓ0 regularizer
• discretized sets D and A
• inherent bilinearity of the model!

We propose an approach based on the Alternating Direction Method of Multipliers (ADMM).

Background and Motivation Problem Statement Error Bounds Algorithmic Approach Experimental Results Acknowledgments

A General-Purpose ADMM-based Approach

We form the augmented Lagrangian L(D, A, X, Λ) = −

• i,j

si,j logpXi,j (Yi,j) + IX (X) + ID(D) + IA(A) + λA0 +tr (Λ(X − DA)) + ρ 2 X − DA2

F,

where Λ is Lagrange multiplier for the equality constraint and ρ > 0 is a parameter, and solve: (S1 :) Xk+1 := min

X∈Rn1×n2 L(Dk, Ak, X, Λk)

(S2 :) Ak+1 := min

A∈Rr×n2 L(Dk, A, Xk+1, Λk)

(S3 :) Dk+1 := min

D∈Rn1×r L(D, Ak+1, Xk+1, Λk)

(S4 :) Λk+1 = Λk + ρ(Xk+1 − Dk+1Ak+1).

Background and Motivation Problem Statement Error Bounds Algorithmic Approach Experimental Results Acknowledgments

Efficiently Solvable Subproblems

We relax D, A, X to closed convex sets, and solve S1-S4 iteratively, as follows... Step S1: After simplification, the solution can be written in terms of scalar prox functions: Xk+1

i,j

= arg min

Xi,j ∈R −si,j logpXi,j (Yi,j) + IX (Xi,j) + ρ

2

• Xi,j − (DkAk)i,j + (Λk)i,j/ρ

2

• prox−si,j logp· (Yi,j )+IX (·)
• (DkAk)i,j − (Λk)i,j/ρ
• .

(Closed-form for three of our examples; use Newton’s Method for the one-bit model w/probit or logit link.)

Background and Motivation Problem Statement Error Bounds Algorithmic Approach Experimental Results Acknowledgments

Efficiently Solvable Subproblems

We relax D, A, X to closed convex sets, and solve S1-S4 iteratively, as follows... Step S1: After simplification, the solution can be written in terms of scalar prox functions: Xk+1

i,j

= arg min

Xi,j ∈R −si,j logpXi,j (Yi,j) + IX (Xi,j) + ρ

2

• Xi,j − (DkAk)i,j + (Λk)i,j/ρ

2

• prox−si,j logp· (Yi,j )+IX (·)
• (DkAk)i,j − (Λk)i,j/ρ
• .

(Closed-form for three of our examples; use Newton’s Method for the one-bit model w/probit or logit link.)

Step S2: The subproblem takes the form min

A∈Rn1×r

IA(A) + λA0 + ρ 2 Xk+1 − DkA + Λk/ρ2

F .

(Solved via “majorization-minimization;” Iterative Hard Thresholding (Blumensath & Davies 2008).)

Background and Motivation Problem Statement Error Bounds Algorithmic Approach Experimental Results Acknowledgments

Efficiently Solvable Subproblems

We relax D, A, X to closed convex sets, and solve S1-S4 iteratively, as follows... Step S1: After simplification, the solution can be written in terms of scalar prox functions: Xk+1

i,j

= arg min

Xi,j ∈R −si,j logpXi,j (Yi,j) + IX (Xi,j) + ρ

2

• Xi,j − (DkAk)i,j + (Λk)i,j/ρ

2

• prox−si,j logp· (Yi,j )+IX (·)
• (DkAk)i,j − (Λk)i,j/ρ
• .

(Closed-form for three of our examples; use Newton’s Method for the one-bit model w/probit or logit link.)

Step S2: The subproblem takes the form min

A∈Rn1×r

IA(A) + λA0 + ρ 2 Xk+1 − DkA + Λk/ρ2

F .

(Solved via “majorization-minimization;” Iterative Hard Thresholding (Blumensath & Davies 2008).)

Step S3: The subproblem takes the form min

D∈Rr×n2

ID(D) + ρ 2 Xk+1 − DAk+1 + Λkρ2

F .

(Efficiently solved via Newton’s Method or closed-form.)

Background and Motivation Problem Statement Error Bounds Algorithmic Approach Experimental Results Acknowledgments

Section 5 Experimental Results

Background and Motivation Problem Statement Error Bounds Algorithmic Approach Experimental Results Acknowledgments

A Comparison with Synthetic Data

Preliminary Experimental Results: We evaluated each of these methods on matrices of size 100 × 1000 with r = 20 and 4 nonzero elements per column of A∗, for varying sampling rates (and different likelihood models). For each, we evaluated the average (over 5 trials) normalized reconstruction error as a function of the sampling rate. Gaussian and Laplace Noises have same variances. For sampling rates > 10−4 ≈ 40%, the error exhibits predicted decay (slope of ≈-1 on the log-log scale).

−1 −0.8 −0.6 −0.4 −0.2 −2 −1 1 2 3

log10(γ) log10

• E

X−X ∗2

F

n 1n 2

• Gaussian

Laplace Poisson

Background and Motivation Problem Statement Error Bounds Algorithmic Approach Experimental Results Acknowledgments

Imaging Example – Gaussian Noise

Original 512 × 512 image reshaped into 256 × 1024 matrix (0.005 ≤ X ∗

i,j ≤ 1.05 for all i, j)

Inner dimension r = 25, noise standard deviation: σ = 0.01, sampling rate = 50% Original Image Samples Estimated Image Estimated A

Background and Motivation Problem Statement Error Bounds Algorithmic Approach Experimental Results Acknowledgments

Imaging Example – Laplace Noise

Original 512 × 512 image reshaped into 256 × 1024 matrix (0.005 ≤ X ∗

i,j ≤ 1.05 for all i, j)

Inner dimension r = 25, noise standard deviation: √ 2/τ = 0.01, sampling rate = 50% Original Image Samples Estimated Image Estimated A

Background and Motivation Problem Statement Error Bounds Algorithmic Approach Experimental Results Acknowledgments

Imaging Example – Poisson-distributed Observations

Original 512 × 512 image reshaped into 256 × 1024 matrix (0.005 ≤ X ∗

i,j ≤ 1.05 for all i, j)

Inner dimension r = 25, sampling rate = 50% Original Image Samples Estimated Image Estimated A

Background and Motivation Problem Statement Error Bounds Algorithmic Approach Experimental Results Acknowledgments

Imaging Example – One-bit Observations

Original 512 × 512 image reshaped into 256 × 1024 matrix (0.005 ≤ X ∗

i,j ≤ 1.05 for all i, j)

Inner dimension r = 25, sampling rate = 50% Original Image Samples Estimated Image Estimated A

Background and Motivation Problem Statement Error Bounds Algorithmic Approach Experimental Results Acknowledgments

Section 6 Acknowledgments

Background and Motivation Problem Statement Error Bounds Algorithmic Approach Experimental Results Acknowledgments

Acknowledgments Collaborators/Co-authors:

Akshay Soni Swayambhoo Jain

• Prof. Stefano Gonella

(UMN ECE PhD Student) (UMN ECE PhD Student) (UMN Civil Engr.)

Background and Motivation Problem Statement Error Bounds Algorithmic Approach Experimental Results Acknowledgments

Acknowledgments Collaborators/Co-authors:

Akshay Soni Swayambhoo Jain

• Prof. Stefano Gonella

(UMN ECE PhD Student) (UMN ECE PhD Student) (UMN Civil Engr.)

Research Support:

NSF EARS (Enhancing Access to the Radio Spectrum) Program DARPA Young Faculty Award

Background and Motivation Problem Statement Error Bounds Algorithmic Approach Experimental Results Acknowledgments

Acknowledgments Collaborators/Co-authors:

Akshay Soni Swayambhoo Jain

• Prof. Stefano Gonella

(UMN ECE PhD Student) (UMN ECE PhD Student) (UMN Civil Engr.)

Research Support:

NSF EARS (Enhancing Access to the Radio Spectrum) Program DARPA Young Faculty Award

Thanks! jdhaupt@umn.edu www.ece.umn.edu/~jdhaupt

(Special thanks to Prof. Julian Wolfson, UMN Dept. of Biostatistics, for the Beamer Template!)