Stable super-resolution limit and smallest singular value of restricted Fourier matrices

Super-resolution refers to the process of recovering the locations and amplitudes of a collection of point sources, represented as a discrete measure, given $M+1$ of its noisy low-frequency Fourier coefficients. The recovery process is highly sensitive to noise whenever the distance $\Delta$ between the two closest point sources is less than $1/M$. This paper studies the {\it fundamental difficulty of super-resolution} and the {\it performance guarantees of a subspace method called MUSIC} in the regime that $\Delta<1/M$. The most important quantity in our theory is the minimum singular value of the Vandermonde matrix whose nodes are specified by the source locations. Under the assumption that the nodes are closely spaced within several well-separated clumps, we derive a sharp and non-asymptotic lower bound for this quantity. Our estimate is given as a weighted $\ell^2$ sum, where each term only depends on the configuration of each individual clump. This implies that, as the noise increases, the super-resolution capability of MUSIC degrades according to a power law where the exponent depends on the cardinality of the largest clump. Numerical experiments validate our theoretical bounds for the minimum singular value and the resolution limit of MUSIC. When there are $S$ point sources located on a grid with spacing $1/N$, the fundamental difficulty of super-resolution can be quantitatively characterized by a min-max error, which is the reconstruction error incurred by the best possible algorithm in the worst-case scenario. We show that the min-max error is closely related to the minimum singular value of Vandermonde matrices, and we provide a non-asymptotic and sharp estimate for the min-max error, where the dominant term is $(N/M)^{2S-1}$.Comment: 47 pages, 8 figure

Characterization of the ammonium transporter family AMF1 in maize

Ammonium is an important inorganic nitrogen source utilized by plants. The uptake of ammonium by plant roots involves two types of ammonium transport pathways, the high-affinity ammonium transporters (HATS) and the low-affinity ammonium transporters (LATS). Genes encoding HATS proteins have been studied well in plants while the LATS pathway is not sufficiently understood. Recently, through a transcriptional linkage to a membrane bound transcription factor, bHLHm1, a novel family of ammonium transporters, AMF1 (ammonium facilitator 1) were discovered. Heterologous expression of yeast and soybean AMF1 orthologs in Xenopus laevis oocytes, indicates AMF1 proteins increase the level of low-affinity ammonium transport. In this study the functional activity of two orthologs of ScAMF1 in maize (ZmAMF1;1 and ZmAMF1;2) were investigated. This study has provided an initial understanding of AMF1’s function in maize through the following outcomes: 1) In yeast, both ZmAMF1;1 and ZmAMF1;2 were capable to rescue growth of a K+-transport mutant CY162 when external ammonium concentration was high but potassium supply was limiting. It is predicted that ZmAMF1;1 and ZmAMF1;2 act as putative NH4+/ H+ antiporters in yeast to facilitate ammonium efflux in the vacuole to the cytoplasm, but the mechanism behind this remains unclear. 2) In maize, loss of ZmAMF1;1 and ZmAMF1;2 activity resulted in a significant increase in short-term ammonium uptake, increased shoot development and root nitrogen content. 3) ZmAMF1;1 and ZmAMF1;2 were both localized in the vascular cylinder in maize roots starved of nitrogen. Both genes were significantly upregulated (around 3.3 to 3.7-fold) in shoot tissues under nitrogen starvation but not downregulated with nitrogen resupply (4 hrs). 4) ZmAMF1;1 and ZmAMF1;2 had different expression profiles in maize where ZmAMF1;1 was more expressed in shoot tissues especially during the reproductive stage, while ZmAMF1;2 expression was primarily in the roots

Characterization of the ammonium transporter family AMF1 in maize

Ammonium is an important inorganic nitrogen source utilized by plants. The uptake of ammonium by plant roots involves two types of ammonium transport pathways, the high-affinity ammonium transporters (HATS) and the low-affinity ammonium transporters (LATS). Genes encoding HATS proteins have been studied well in plants while the LATS pathway is not sufficiently understood. Recently, through a transcriptional linkage to a membrane bound transcription factor, bHLHm1, a novel family of ammonium transporters, AMF1 (ammonium facilitator 1) were discovered. Heterologous expression of yeast and soybean AMF1 orthologs in Xenopus laevis oocytes, indicates AMF1 proteins increase the level of low-affinity ammonium transport. In this study the functional activity of two orthologs of ScAMF1 in maize (ZmAMF1;1 and ZmAMF1;2) were investigated. This study has provided an initial understanding of AMF1’s function in maize through the following outcomes: 1) In yeast, both ZmAMF1;1 and ZmAMF1;2 were capable to rescue growth of a K+-transport mutant CY162 when external ammonium concentration was high but potassium supply was limiting. It is predicted that ZmAMF1;1 and ZmAMF1;2 act as putative NH4+/ H+ antiporters in yeast to facilitate ammonium efflux in the vacuole to the cytoplasm, but the mechanism behind this remains unclear. 2) In maize, loss of ZmAMF1;1 and ZmAMF1;2 activity resulted in a significant increase in short-term ammonium uptake, increased shoot development and root nitrogen content. 3) ZmAMF1;1 and ZmAMF1;2 were both localized in the vascular cylinder in maize roots starved of nitrogen. Both genes were significantly upregulated (around 3.3 to 3.7-fold) in shoot tissues under nitrogen starvation but not downregulated with nitrogen resupply (4 hrs). 4) ZmAMF1;1 and ZmAMF1;2 had different expression profiles in maize where ZmAMF1;1 was more expressed in shoot tissues especially during the reproductive stage, while ZmAMF1;2 expression was primarily in the roots