Download
1 / 50
500 likes | 616 Vues
This lecture provides an in-depth exploration of polynomial embeddings for solving quadratic algebraic equations. It discusses the invertibility of nonlinear transformations, explores the challenges of reconstructing data from magnitude information, and presents various algebraic approaches including quasi-linear and hierarchical embeddings. Theoretical bounds such as the Cramér-Rao Lower Bound (CRLB) and performance analysis of modified least square estimators are also examined. Applications in fields like X-ray crystallography and speech processing highlight the relevance of these mathematical concepts.
E N D
Polynomial Embeddings for Quadratic Algebraic Equations Radu Balan University of Maryland, College Park, MD 20742 Math-CS Joint Lecture, Drexel University Monday April 23, 2012
Overview • Introduction: Motivation and statement of problem • Invertibility Results in the Real Case • The Algebraic Approach: • Quasi-Linear Embeddings • Hierarchical Embeddings • Numerical Analysis • Modified Least Square Estimator • Theoretical Bounds: CRLB • Performance Analysis
1. Introduction: Motivation Inversion of Nonlinear Transformations y = A x ? knowns BA=Identity Question:What if |y| is known instead (that is, one looses the phase information) ? Where is important: X-Ray Crystallography, Speech Processing
1. Introduction: Statement of the Problem Reconstruction from magnitudes of frame coefficients a complete set of vectors (frame) for the n-dimensional Hilbert space H (H=Cn or H=Rn). Equivalence relation: x,yH, x~yiff there is a scalar z, |z|=1 so that x=zy (real case: x=y ; complex case: x=eiy). Let . Define Problems: Is an injective map? If it is, how to invert it efficiently?
H f1 x (x) fm f2 Rm
2. Invertibility Results: Real Case (1) • Real Case:K=R • Theorem[R.B.,Casazza, Edidin, ACHA(2006)] • is injective iff for any subset JF either J or F\J spans Rn. • Corollaries [2006] • if m 2n-1, and a generic frame set F, then is injective; • if m2n-2 then for any set F, cannot be injective; • if any n-element subset of F is linearly independent, then is injective; for m=2n-1 this is a necessary and sufficient condition.
Invertibility Results: Real Case (2) • Real Case:K=R • Theorem [R.B.(2012)] is injective iff any one of the following equivalent conditions holds true: • For any x,yRn, x≠0, y≠0, • There is a constant a>0 so that for all x, • RI
Invertibility Results: Real Case (3) complete set of vectors (frame) for H=Rn. One would expect that if is injective and m>2n-1 then there is a strict subset J{1,2,…,m} so that π is injective, where π:RmR|J| is the restriction to J index. However the next example shows this is not the case. Example. Consider n=3, m=6, and F the set of columns of the following matrix F = Note that for any subset J of 3 columns, either J or F\J is linearly dependent. Thus is injective but removing any column makes πnot injective.
3. The Algebraic Approach 3.1 Quasi-Linear Embeddings Example (a) Consider the real case: n=3 , m=6. Frame vectors: Need to solve a system of the form:
Then factor: Thus, we obtain:
Example (b) Consider the real case: n=3 , m=5. Frame vectors: Need to solve the system: Is it possible? How? X
Let’s square again: We obtained 5 linear equations with 6 unknown monomials. Idea: Let’s multiply again these equations (square and cross) New equations: : 15 equations New variables (monomials): : 15 unknowns
3.2 Hierarchical Embeddings Primary data: Level d embedding: , Identify: Then a homogeneous polynomial of total degree 2d.
How many monomials? Real case: number of monomials of degree 2d in n variables: Complex case: … Number of degree (d,d) in n variables: Define redundancy at level d: (R.B. [SampTA2009])
Fundamental question: How many equations are linearly independent? Recall: , Note: and The matrix is not canonical, and so is * for d>1. However its range is basis independent. We are going to compute this range in terms of a canonical matrix.
TheoremThe following hold true: (as a quadratic form) Rows of are linearly independent iff where is the mdxmd matrix given by Real case: Complex case:
Let denote the Gram matrix And for integer p. Theorem In either real or complex case: Hence for d=1, the number of independent quadratics is given by: TheoremFor d=2, Remark Note the k1=k2,l1=l2submatrix of is
3.3. Numerical analysis Results for the complex case: random frames n=3,m=6
n=16 m=64
n=32 m=128
n=4 m=16
n=4 m=14 Note 6 zero eigenvalues of instead of 5.
n=4 m=15 Number of zero eigenvalues = 20 =120-100, as expected.
4. Modified Least Square Error Estimator • Model: “Vanilla” Least-Square-Estimator (LSE): We modify this criterion in two ways: Replace X by a rank r positive matrix Y Regularize the criterion by adding a norm of Y
Optimization procedure Our approach: Start with a large and decrease its value over time; Replace one L in the inner quadratic term by a previous estimate Penalize large successive variations.
Algorithm (Part I) Step I: Initialization Step II: Iteration Step III: Factorization How to initialize? How to adapt? Convergence?
Initialization Set:
Convergence Consider the iterative process: Theorem Assume (t)t,(µt)t are monotonically decreasing non-negative sequences. Then (jt)t≥0 is a monotonically decreasing convergent sequence.
The LSE/MLE is a biased estimator. Modified CRLB for biased estimators: Asymptotically (for high SNR):
6. Performance Analysis Mintrace algorithm: Candes, Strohmer, Voroninski (2011) n=3 , m=9 , d = 1 (dlevel) , r = 2 and , decrease by 5% every step w/ saturation (subspace)
