Algebraic compressed sensing

Question

1. What is the dimension of the model X in algebraic compressed sensing?

2. What lies beyond the restricted isometry property?

3. How much does the reconstruction move in compressed sensing algorithms?

4. What does existence in algebraic compressed sensing refer to?

Accepted Answer

The dimension of the model X, denoted as d, is introduced formally in Section 4.2. It measures the 'intrinsic size' of the model X and coincides with the dimension of the set of non-singular points of X, which is a smooth manifold embedded in R^n [14]. In the explicit setting (ES), the dimension of X is the maximal rank of the Jacobian of the parametrization ph at points in R^n [31]. The dimension of the model X determines the required number of measurements s for the inverse problem to be well-posed. The dimension d is fixed, while the number of measurements s can vary, reflecting the design choice of the measurement map. The dimension of the model X plays a crucial role in the well-posedness of algebraic compressed sensing problems.

Accepted Answer

The -restricted isometry property ( -RIP) of the measurement map u : X - k s [28, 29, 32] , where k = R or k = C, is a core concept in compressed sensing. Maps satisfying the RIP are a special type of global bi-Lipschitz embeddings. However, demanding that the measurement map is globally bi-Lipschitz can be challenging to verify theoretically and imposes severe geometric constraints on X and u. Relaxing the requirement of bi-Lipschitz embeddability can often lead to recoverability and identifiability with a smaller number of measurements. For a generic linear map, d measurements suffice for generic recoverability and d + 1 measurements suffice for generic identifiability. This observation is not new and has been proven in various studies. Therefore, beyond the restricted isometry property, there are alternative approaches to achieve recoverability and identifiability with fewer measurements, especially for linear maps.

Accepted Answer

The reconstruction error in compressed sensing algorithms is quantified using the condition number, which measures the amplification of infinitesimal perturbations in the output of a function. A forward stable algorithm has a reconstruction error bounded by the condition number multiplied by a constant K, which quantifies the amplification of the inherent sensitivity of the problem. The condition number is computed from the singular values of the derivative of the measurement map, making it easier to approximate numerically. This approach sacrifices a global upper bound on the condition number but provides local upper bounds and identifiability for specific problem instances.

Accepted Answer

Existence in algebraic compressed sensing refers to deciding whether a given y k s sits in the image of the model X under the measurement map u. It determines if a solution exists for a specific data point in the compressed sensing problem. This concept is crucial in understanding the feasibility of recovering signals from limited measurements. The existence of solutions is formulated for 'generic' data points, which form an open dense subset of all points. This ensures that the analysis is applicable to a wide range of scenarios and provides a comprehensive understanding of the problem's solvability.

Accepted Answer

For a given point y = (y1, ..., ys), solutions exist in polynomial systems if the following conditions are met. First, if X is the zero locus of polynomials f1, ..., fk, then the system of k+s polynomial equations f1(x) = *, ..., fk(x) = y1 - u1(x), ..., ys - us(x) = 0 has a solution. Second, if X is the image of a polynomial parameterization, then the system of s polynomial equations y1 - u1(ph1(t), ..., phn(t)) = *, ..., ys - us(ph1(t), ..., phn(t)) = 0 has a solution. The existence of solutions is characterized by the ideal g1, ..., gN not containing 1, which can be tested using Grobner basis methods. For k = C, a finite number of polynomials h1(y), ..., hM determine the existence of solutions. For k = R, polynomial equations and inequalities need to be checked. Deciding the existence of solutions is an NP-hard problem, but numerical methods like certified homotopy continuation can be used for practical purposes. A map is recoverable at a point xX if the fiber u-1(u(x)) is finite, and a map is everywhere or generically recoverable if it is recoverable at every point or generic points, respectively.

Accepted Answer

A generic affine linear map is everywhere recoverable if the number of measurements is s >= d. If k = C and s = d, the number of solutions equals the degree of X. In the real case, the number of solutions is bounded by d. This applies to certain polynomial maps as well. The degree of X is a constant d, as discussed in Section 4.3. The number of real solutions for a generic problem instance is also bounded by d. These conditions ensure the recoverability of a generic affine linear map.

Accepted Answer

A measurement map defined by generic polynomials of degree p is recoverable if s >= d. For k = C and s = d, the number of solutions equals d^2p. If k = R, the number of real solutions is bounded by d^2p. The answer applies to generic polynomial maps with components of different degrees. A proof can be adapted from Noether's Normalization Lemma. Randomly sampled maps are recoverable with probability 1. Low-rank matrix recovery from Gaussian measurements requires 10r(m+n) measurements. Coordinate projections may not follow the conditions, but have specific results.

Accepted Answer

A map u : X - k s is generically identifiable if s >= d + 1. When k = C, d > 0, and p > 1, the condition s >= d + 1 is necessary as well. This means that for a map defined by s generic polynomials of degree p, it will be generically identifiable if the value of s is greater than or equal to the sum of dimension d and 1. This condition ensures that the map has enough degrees of freedom to be identifiable in a generic sense. The condition also takes into account the specific case when k = C, d > 0, and p > 1, which further emphasizes the importance of meeting the s >= d + 1 requirement for generically identifiable maps.

Accepted Answer

The conditions under which the measurement map is locally or globally injective include the existence of a (local) inverse map that is at least continuous. For an inverse problem to be well-posed, the (local) inverse map must be smooth, continuous, and locally Lipschitz continuous. A global diffeomorphism onto its image requires the map to be smooth, injective, and have a smooth inverse. A local diffeomorphism onto its image at a point requires a Euclidean open neighborhood where the restriction of the map is a global diffeomorphism onto its image, and the image of the neighborhood is open in the function's image.

Accepted Answer

The condition number with respect to the Euclidean metrics on both domain and codomain at a smooth point x X is k(x) = 1 s d (dxu), where s d (d x u) is the dth largest singular value of the derivative of u : X - k s at x. More concretely, let M k sxn be the matrix representing the linear map m : k n - k s in the ambient space. Let Q k nxd be a matrix whose columns form an orthonormal basis of the tangent space T x X. Then k(x) = 1 s d (M Q).

Accepted Answer

In classic algebraic geometry, an algebraic variety is a subset X defined as the vanishing set of a set of polynomials f1, ..., fk in n variables x1, ..., xn with coefficients in k. It is denoted as Ik(X) := {fk[x1, ..., xn] : f(x) = 0 for all x in X}. The variety can be real or complex, depending on whether k = R or k = C. It is a fundamental concept used in proving the correctness of answers in algebraic geometry.

Accepted Answer

Numerical homotopy continuation methods are numerical techniques used to compute all isolated solutions of a system of polynomial equations. They involve creating a homotopy, which is a continuous transformation from one system of equations to another, and then tracking the solutions as the homotopy parameter changes. These methods are particularly useful for solving small algebraic compressed sensing problem instances. The HomotopyContinuation.jl library in Julia provides tools for implementing these methods. By using homotopy continuation, researchers can find alternative decompositions into rank-1 terms, depending on the measurement map u. This process requires a parameterization ph from a Euclidean space of dimension d into T 2, where T 2 is the set of tensors in R 4 R 3 R 2 R 4x3x2 of rank bounded by 2. However, it is known that this parameterization is not always possible, and T 2 needs to be covered with multiple charts R d - T 2. The experiments conducted in the numerical experiment section confirm the effectiveness of homotopy continuation methods in solving algebraic compressed sensing problems.

Algebraic compressed sensing

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Most frequently asked questions

1. What is the dimension of the model X in algebraic compressed sensing?

2. What lies beyond the restricted isometry property?

3. How much does the reconstruction move in compressed sensing algorithms?

4. What does existence in algebraic compressed sensing refer to?

5. What conditions ensure solutions for a given point in polynomial systems?

6. Under what conditions is a generic affine linear map recoverable?

7. Recoverable conditions for measurement map u:X-k s?

8. Under what conditions is a map u : X - k s defined by s generic polynomials of degree p generically identifiable?

9. What conditions ensure the measurement map is locally or globally injective? What are the requirements for an inverse problem to be well-posed?

10. How does one compute the condition number of the inverse problem (IP) at a smooth point x X?

11. What is an algebraic variety in classic algebraic geometry?

12. How do numerical homotopy continuation methods work?

References

Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information

Tensor Decompositions and Applications

Near-Optimal Signal Recovery From Random Projections: Universal Encoding Strategies?

Decoding by linear programming

Stable signal recovery from incomplete and inaccurate measurements

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