An algebraic characterization of injectivity in phase retrieval (1312.0158v1)

Abstract: A complex frame is a collection of vectors that span $\mathbb{C}^M$ and define measurements, called intensity measurements, on vectors in $\mathbb{C}^M$. In purely mathematical terms, the problem of phase retrieval is to recover a complex vector from its intensity measurements, namely the modulus of its inner product with these frame vectors. We show that any vector is uniquely determined (up to a global phase factor) from $4M-4$ generic measurements. To prove this, we identify the set of frames defining non-injective measurements with the projection of a real variety and bound its dimension.

• The paper establishes that generic frames with 4M-4 vectors ensure injectivity in phase retrieval.
• It employs algebraic geometry to rigorously characterize non-injective cases in complex signal recovery.
• The findings have practical implications for imaging and quantum state tomography by ensuring unique reconstruction.

An Algebraic Characterization of Injectivity in Phase Retrieval

The paper investigates phase retrieval in signal processing through an algebraic approach. Specifically, it addresses the problem of recovering a complex vector in a finite-dimensional complex space, $\mathbb{C}^M$, from intensity measurements. These measurements are expressed as the moduli squared of the inner products between the vector and a frame, a spanning set of vectors in $\mathbb{C}^M$. The challenge arises because the phase information, crucial for perfect reconstruction, is obscured in such a non-linear measurement process.

Mathematical Framing of the Problem

The mathematical framework situates the problem within the context of real and complex algebraic geometry. This is instrumental in handling the inherently non-linear transformations involved in phase retrieval. The phase retrieval process can be represented through a non-linear mapping, $$A_\Phi: (\mathbb{C}^M/S^1) \to (\mathbb{R}_{\geq 0})^N$$, which must be injective to ensure unique signal recovery, modulo a global phase factor. The notation $(\mathbb{C}^M/S^1)$ designates equivalence classes of complex vectors, where each class comprises vectors differing by a multiplicative scalar with unit magnitude.

Main Results and Conjecture

The authors establish that for a generic frame, composed of $4M-4$ vectors in $\mathbb{C}^M$, the mapping $A_\Phi$ is injective. This conclusion supports part (b) of the “$4M-4$ Conjecture”. The conjecture asserts two parts: (a) for $N < 4M-4$, $A_\Phi$ is not injective, and (b) for $N \geq 4M-4$, injectivity is achieved for generic frames.

As the paper builds up to its main theorem, it leverages results from algebraic geometry, including concepts like the dimension of algebraic sets and the properties of Zariski open sets. These tools help in characterizing when the viewed problem's system of equations reaches injectivity.

Proof Techniques

The proof of the main theorem involves characterizing the set of non-injective frames. The authors associate the non-injectivity of $A_\Phi$ with a corresponding algebraic structure, a real variety in the project's coordinates. They then frame this variety's projection within a real Zariski open set context to identify an open dense set of injective frames.

Implications and Further Research

The findings have practical significance in fields where signal processing via phase retrieval is deployed, such as coherent diffractive imaging and quantum state tomography. Theoretically, the approach used in this paper integrates algebraic geometry with signal processing, offering a rigorous basis for assessing signal retrieval methods' capabilities and limitations.

Moreover, the open case—the exploration of real algebraic geometry applied to minimal measurements—is identified as an area ripe for further investigation. Particularly, confirming part (a) of the $4M-4$ Conjecture across broader contexts than those outlined, like specific matrix dimensions that are not handled due to symmetry properties, remains a stimulating challenge.

Conclusion

This paper significantly advances the algebraic understanding of phase retrieval, demonstrating that algebraic geometry provides an adept framework for resolving injectivity questions in non-linear measurement maps. As this work establishes robust results within prescribed dimensions, it offers pathways for extending algebraic techniques to other domains where phase information retrieval forms the backbone of real-world applications.