Phase retrieval without small-ball probability assumptions: Stability and uniqueness

2015 ◽  
Author(s):  
Felix Krahmer ◽  
Yi-Kai Liu
Keyword(s):  
Phase Retrieval ◽  
Small Ball Probability ◽  
Small Ball

scholarly journals Complex Phase Retrieval from Subgaussian Measurements

2020 ◽  
Vol 26 (6) ◽  
Author(s):  
Felix Krahmer ◽  
Dominik Stöger
Keyword(s):  
Large Class ◽  
Recent Work ◽  
Additional Assumption ◽  
Phase Retrieval ◽  
Small Ball Probability ◽  
Complex Phase ◽  
Small Ball ◽  
Proof Strategy ◽  
Complex Valued ◽  
Unknown Vector

AbstractPhase retrieval refers to the problem of reconstructing an unknown vector $$x_0 \in {\mathbb {C}}^n$$ x 0 ∈ C n or $$x_0 \in {\mathbb {R}}^n $$ x 0 ∈ R n from m measurements of the form $$y_i = \big \vert \langle \xi ^{\left( i\right) }, x_0 \rangle \big \vert ^2 $$ y i = | ⟨ ξ i , x 0 ⟩ | 2 , where $$ \left\{ \xi ^{\left( i\right) } \right\} ^m_{i=1} \subset {\mathbb {C}}^m $$ ξ i i = 1 m ⊂ C m are known measurement vectors. While Gaussian measurements allow for recovery of arbitrary signals provided the number of measurements scales at least linearly in the number of dimensions, it has been shown that ambiguities may arise for certain other classes of measurements $$ \left\{ \xi ^{\left( i\right) } \right\} ^{m}_{i=1}$$ ξ i i = 1 m such as Bernoulli measurements or Fourier measurements. In this paper, we will prove that even when a subgaussian vector $$ \xi ^{\left( i\right) } \in {\mathbb {C}}^m $$ ξ i ∈ C m does not fulfill a small-ball probability assumption, the PhaseLift method is still able to reconstruct a large class of signals $$x_0 \in {\mathbb {R}}^n$$ x 0 ∈ R n from the measurements. This extends recent work by Krahmer and Liu from the real-valued to the complex-valued case. However, our proof strategy is quite different and we expect some of the new proof ideas to be useful in several other measurement scenarios as well. We then extend our results $$x_0 \in {\mathbb {C}}^n $$ x 0 ∈ C n up to an additional assumption which, as we show, is necessary.

scholarly journals Small ball probability estimates in terms of width

2005 ◽  
Vol 169 (3) ◽  
pp. 305-314 ◽  
Author(s):  
Rafał Latała ◽  
Krzysztof Oleszkiewicz
Keyword(s):  
Small Ball Probability ◽  
Small Ball ◽  
Probability Estimates

scholarly journals On the Lower Classes of Some Mixed Fractional Gaussian Processes with Two Logarithmic Factors

2008 ◽  
Vol 2008 ◽  
pp. 1-16 ◽  
Author(s):  
Charles El-Nouty
Keyword(s):  
General Result ◽  
Gaussian Processes ◽  
Brownian Sheet ◽  
Random Vectors ◽  
Fractional Brownian Sheet ◽  
Integral Test ◽  
Small Ball Probability ◽  
Small Ball ◽  
Sufficiency Part ◽  
Necessity Part

We introduce the fractional mixed fractional Brownian sheet and investigate the small ball behavior of its sup-norm statistic by establishing a general result on the small ball probability of the sum of two not necessarily independent joint Gaussian random vectors. Then, we state general conditions and characterize the sufficiency part of the lower classes of some statistics of the above process by an integral test. Finally, when we consider the sup-norm statistic, the necessity part is given by a second integral test.

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