Self-Calibration and Biconvex Compressive Sensing Shuyang Ling Department of Mathematics, UC Davis July 12, 2017 Shuyang Ling (UC Davis) SIAM Annual Meeting, 2017, Pittsburgh July 12, 2017 1 / 22
Acknowledgements Research in collaboration with: Prof.Thomas Strohmer (UC Davis) This work is sponsored by NSF-DMS and DARPA. Shuyang Ling (UC Davis) SIAM Annual Meeting, 2017, Pittsburgh July 12, 2017 2 / 22
Outline (a) Self-calibration and mathematical framework (b) Biconvex compressive sensing in array signal processing (c) SparseLift: a convex approach towards biconvex compressive sensing (d) Theory and numerical simulations Shuyang Ling (UC Davis) SIAM Annual Meeting, 2017, Pittsburgh July 12, 2017 3 / 22
Calibration Calibration: Calibration is to adjust one device with the standard one. Why? To reduce or eliminate bias and inaccuracy. Difficult or even impossible to calibrate high-performance hardware. Self-calibration: Equip sensors with a smart algorithm which takes care of calibration automatically. Shuyang Ling (UC Davis) SIAM Annual Meeting, 2017, Pittsburgh July 12, 2017 4 / 22
Calibration Calibration: Calibration is to adjust one device with the standard one. Why? To reduce or eliminate bias and inaccuracy. Difficult or even impossible to calibrate high-performance hardware. Self-calibration: Equip sensors with a smart algorithm which takes care of calibration automatically. Shuyang Ling (UC Davis) SIAM Annual Meeting, 2017, Pittsburgh July 12, 2017 4 / 22
Calibration realized by machine? Uncalibrated device leads to imperfect sensing We encounter imperfect sensing all the time: the sensing matrix A ( h ) depending on an unknown calibration parameter h , y = A ( h ) x + w . This is too general to solve for h and x jointly. Examples: Phase retrieval problem: h is the unknown phase of the Fourier transform of x . Cryo-electron microscopy images: h can be the unknown orientation of a protein molecule and x is the particle. Shuyang Ling (UC Davis) SIAM Annual Meeting, 2017, Pittsburgh July 12, 2017 5 / 22
Calibration realized by machine? Uncalibrated device leads to imperfect sensing We encounter imperfect sensing all the time: the sensing matrix A ( h ) depending on an unknown calibration parameter h , y = A ( h ) x + w . This is too general to solve for h and x jointly. Examples: Phase retrieval problem: h is the unknown phase of the Fourier transform of x . Cryo-electron microscopy images: h can be the unknown orientation of a protein molecule and x is the particle. Shuyang Ling (UC Davis) SIAM Annual Meeting, 2017, Pittsburgh July 12, 2017 5 / 22
A simplified but important model Our focus: One special case is to assume A ( h ) to be of the form A ( h ) = D ( h ) A where D ( h ) is an unknown diagonal matrix. However, this seemingly simple model is very useful and mathematically nontrivial to analyze. Phase and gain calibration in array signal processing Blind deconvolution (image deblurring; joint channel and signal estimation, etc.) Shuyang Ling (UC Davis) SIAM Annual Meeting, 2017, Pittsburgh July 12, 2017 6 / 22
A simplified but important model Our focus: One special case is to assume A ( h ) to be of the form A ( h ) = D ( h ) A where D ( h ) is an unknown diagonal matrix. However, this seemingly simple model is very useful and mathematically nontrivial to analyze. Phase and gain calibration in array signal processing Blind deconvolution (image deblurring; joint channel and signal estimation, etc.) Shuyang Ling (UC Davis) SIAM Annual Meeting, 2017, Pittsburgh July 12, 2017 6 / 22
Self-calibration in array signal processing Calibration in the DOA (direction of arrival estimation) One calibration issue comes from the unknown gains of the antennae caused by temperature or humidity. Consider s signals impinging on an array of L antennae. Antenna elements s 𝜄 " 𝜄 ' � DA (¯ y = θ k ) x k + w k =1 𝜄 # 𝜄 $ where D is an unknown diagonal matrix and d ii is the unknown gain for i -th sensor. A ( θ ): array mani- 𝜄 % 𝜄 & fold. ¯ θ k : unknown direction of ar- rival. { x k } s k =1 are the impinging signals. Shuyang Ling (UC Davis) SIAM Annual Meeting, 2017, Pittsburgh July 12, 2017 7 / 22
Self-calibration in array signal processing Calibration in the DOA (direction of arrival estimation) One calibration issue comes from the unknown gains of the antennae caused by temperature or humidity. Consider s signals impinging on an array of L antennae. Antenna elements s 𝜄 " 𝜄 ' � DA (¯ y = θ k ) x k + w k =1 𝜄 # 𝜄 $ where D is an unknown diagonal matrix and d ii is the unknown gain for i -th sensor. A ( θ ): array mani- 𝜄 % 𝜄 & fold. ¯ θ k : unknown direction of ar- rival. { x k } s k =1 are the impinging signals. Shuyang Ling (UC Davis) SIAM Annual Meeting, 2017, Pittsburgh July 12, 2017 7 / 22
How is it related to compressive sensing? Discretize the manifold function A ( θ ) over [ − π ≤ θ < π ] on N grid points. y = DAx + w where | · · · | ∈ C L × N A = A ( θ 1 ) · · · A ( θ N ) | · · · | To achieve high resolution, we usually have L ≤ N . x ∈ C N × 1 is s -sparse. Its s nonzero entries correspond to the directions of signals. Moreover, we don’t know the locations of nonzero entries. Subspace constraint: assume D = diag( Bh ) where B is a known L × K matrix and K < L . Number of constraints: L ; number of unknowns: K + s . Shuyang Ling (UC Davis) SIAM Annual Meeting, 2017, Pittsburgh July 12, 2017 8 / 22
How is it related to compressive sensing? Discretize the manifold function A ( θ ) over [ − π ≤ θ < π ] on N grid points. y = DAx + w where | · · · | ∈ C L × N A = A ( θ 1 ) · · · A ( θ N ) | · · · | To achieve high resolution, we usually have L ≤ N . x ∈ C N × 1 is s -sparse. Its s nonzero entries correspond to the directions of signals. Moreover, we don’t know the locations of nonzero entries. Subspace constraint: assume D = diag( Bh ) where B is a known L × K matrix and K < L . Number of constraints: L ; number of unknowns: K + s . Shuyang Ling (UC Davis) SIAM Annual Meeting, 2017, Pittsburgh July 12, 2017 8 / 22
Self-calibration and biconvex compressive sensing Goal : Find ( h , x ) s.t. y = diag( Bh ) Ax + w and x is sparse. Biconvex compressive sensing We are solving a biconvex (not convex) optimization problem to recover sparse signal x and calibrating parameter h . h , x � diag( Bh ) Ax − y � 2 + λ � x � 1 min A ∈ C L × N and B ∈ C L × K are known. h ∈ C K × 1 and x ∈ C N × 1 are unknown. x is sparse. Remark: If h is known, x can be recovered; if x is known, we can find h as well. Regarding identifiability issue, See [Lee, etc. 15]. Shuyang Ling (UC Davis) SIAM Annual Meeting, 2017, Pittsburgh July 12, 2017 9 / 22
Biconvex compressive sensing Goal: we want to find h and a sparse x from y , B and A . 𝒛: 𝑀×1 𝑪: 𝑀×𝐿 𝒊: 𝐿×1 𝑦: 𝑂×1, 𝒙: 𝑀×1 𝐵: 𝑀×𝑂 𝑡 -sparse = + ⊙ Unknown parameters Shuyang Ling (UC Davis) SIAM Annual Meeting, 2017, Pittsburgh July 12, 2017 10 / 22
Convex approach and lifting Two-step convex approach (a) Lifting: convert bilinear to linear constraints Widely used in phase retrieval [Cand` es, etc, 11], blind deconvolution [Ahmed, etc, 11], etc... (b) Solving a convex relaxation (semi-definite program) to recover h 0 x ∗ 0 . Shuyang Ling (UC Davis) SIAM Annual Meeting, 2017, Pittsburgh July 12, 2017 11 / 22
Lifting: from bilinear to linear Step 1: lifting Let a i be the i -th column of A ∗ and b i be the i -th column of B ∗ . y i = ( Bh 0 ) i x ∗ 0 a i + w i = b ∗ i h 0 x ∗ 0 a i + w i . 0 and define the linear operator A : C K × N → C L as, Let X 0 := h 0 x ∗ i Za i } L i �} L A ( Z ) := { b ∗ i =1 = {� Z , b i a ∗ i =1 . Then, there holds y = A ( X 0 ) + w . i : C L → C K × N . In this way, A ∗ ( z ) = � L i =1 z i b i a ∗ Shuyang Ling (UC Davis) SIAM Annual Meeting, 2017, Pittsburgh July 12, 2017 12 / 22
Rank-1 matrix recovery Lifting: recovery of a rank - 1 and row-sparse matrix Find Z s.t. rank( Z ) = 1 A ( Z ) = A ( X 0 ) Z has sparse rows 0 , h 0 ∈ C K and x 0 ∈ C N with � X 0 � 0 = Ks where X 0 = h 0 x ∗ � x 0 � 0 = s . 0 0 h 1 x i 1 0 · · · 0 h 1 x i s 0 · · · 0 0 0 0 · · · 0 0 · · · 0 h 2 x i 1 h 2 x i s Z = . . . . . . . . ... ... . . . . . . . . . . . . . . . . 0 0 h K x i 1 0 · · · 0 h K x i s 0 · · · 0 K × N An NP-hard problem to find such a rank-1 and sparse matrix. Shuyang Ling (UC Davis) SIAM Annual Meeting, 2017, Pittsburgh July 12, 2017 13 / 22
Rank-1 matrix recovery Lifting: recovery of a rank - 1 and row-sparse matrix Find Z s.t. rank( Z ) = 1 A ( Z ) = A ( X 0 ) Z has sparse rows 0 , h 0 ∈ C K and x 0 ∈ C N with � X 0 � 0 = Ks where X 0 = h 0 x ∗ � x 0 � 0 = s . 0 0 h 1 x i 1 0 · · · 0 h 1 x i s 0 · · · 0 0 0 0 · · · 0 0 · · · 0 h 2 x i 1 h 2 x i s Z = . . . . . . . . ... ... . . . . . . . . . . . . . . . . 0 0 h K x i 1 0 · · · 0 h K x i s 0 · · · 0 K × N An NP-hard problem to find such a rank-1 and sparse matrix. Shuyang Ling (UC Davis) SIAM Annual Meeting, 2017, Pittsburgh July 12, 2017 13 / 22
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