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1. Super-resolution multi-reference alignment

   https://doi.org/10.1093/imaiai/iaab003  

   Bendory, Tamir; Jaffe, Ariel; Leeb, William; Sharon, Nir; Singer, Amit (February 2021, Information and Inference: A Journal of the IMA)

   null (Ed.)

   Abstract We study super-resolution multi-reference alignment, the problem of estimating a signal from many circularly shifted, down-sampled and noisy observations. We focus on the low SNR regime, and show that a signal in $${\mathbb{R}}^M$$ is uniquely determined when the number $$L$$ of samples per observation is of the order of the square root of the signal’s length ($$L=O(\sqrt{M})$$). Phrased more informally, one can square the resolution. This result holds if the number of observations is proportional to $$1/\textrm{SNR}^3$$. In contrast, with fewer observations recovery is impossible even when the observations are not down-sampled ($L=M$). The analysis combines tools from statistical signal processing and invariant theory. We design an expectation-maximization algorithm and demonstrate that it can super-resolve the signal in challenging SNR regimes. 

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2. Power Spectrum Unbiasing for Dilation-Invariant Multi-reference Alignment

   https://doi.org/10.1007/s00041-023-10023-5  

   Hirn, Matthew; Little, Anna (August 2023, Journal of Fourier Analysis and Applications)

   This article discusses a generalization of the 1-dimensional multi-reference alignment problem. The goal is to recover a hidden signal from many noisy observations, where each noisy observation includes a random translation and random dilation of the hidden signal, as well as high additive noise. We propose a method that recovers the power spectrum of the hidden signal by applying a data-driven, nonlinear unbiasing procedure, and thus the hidden signal is obtained up to an unknown phase. An unbiased estimator of the power spectrum is defined, whose error depends on the sample size and noise levels, and we precisely quantify the convergence rate of the proposed estimator. The unbiasing procedure relies on knowledge of the dilation distribution, and we implement an optimization procedure to learn the dilation variance when this parameter is unknown. Our theoretical work is supported by extensive numerical experiments on a wide range of signals. 

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3. Statistical and Computational Complexities of BFGS Quasi-Newton Method for Generalized Linear Models

   Jin, Qiujiang; Ren, Tongzheng; Ho, Nhat; Mokhtari, Aryan (May 2024, Transactions on machine learning research)

   The gradient descent (GD) method has been used widely to solve parameter estimation in generalized linear models (GLMs), a generalization of linear models when the link function can be non-linear. In GLMs with a polynomial link function, it has been shown that in the high signal-to-noise ratio (SNR) regime, due to the problem's strong convexity and smoothness, GD converges linearly and reaches the final desired accuracy in a logarithmic number of iterations. In contrast, in the low SNR setting, where the problem becomes locally convex, GD converges at a slower rate and requires a polynomial number of iterations to reach the desired accuracy. Even though Newton's method can be used to resolve the flat curvature of the loss functions in the low SNR case, its computational cost is prohibitive in high-dimensional settings as it is $$\mathcal{O}(d^3)$$, where $$d$$ the is the problem dimension. To address the shortcomings of GD and Newton's method, we propose the use of the BFGS quasi-Newton method to solve parameter estimation of the GLMs, which has a per iteration cost of $$\mathcal{O}(d^2)$$. When the SNR is low, for GLMs with a polynomial link function of degree $$p$$, we demonstrate that the iterates of BFGS converge linearly to the optimal solution of the population least-square loss function, and the contraction coefficient of the BFGS algorithm is comparable to that of Newton's method. Moreover, the contraction factor of the linear rate is independent of problem parameters and only depends on the degree of the link function $$p$$. Also, for the empirical loss with $$n$$ samples, we prove that in the low SNR setting of GLMs with a polynomial link function of degree $$p$$, the iterates of BFGS reach a final statistical radius of $$\mathcal{O}((d/n)^{\frac{1}{2p+2}})$$ after at most $$\log(n/d)$$ iterations. This complexity is significantly less than the number required for GD, which scales polynomially with $(n/d)$. 

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4. Low frequency electrochemical noise in AlGaN/GaN field effect transistor biosensors

   https://doi.org/10.1063/5.0014495  

   Paul Bertani, Yuji Wang (July 2020, Applied physics letters)

   Little has been studied on how the electrochemical noise impacts the limit of detection of field effect transistor (FET) biosensors. Herein, we investigate low frequency noise associated with phosphate-buffered saline (PBS) solutions at varying ionic strengths (Ni) under both weak and strong gate biases corresponding to saturation and sub-threshold regimes, respectively, in AlGaN/GaN heterojunction FET biosensors. We show that the electrochemical noise is strongly dependent on the ionic strength and gate biasing conditions. In the saturation regime (low bias), varying the ionic strength (a range of 10−6× PBS to PBS 1 × stock solutions used for testing) has little to no effect on the characteristic frequency exponent 𝛽(𝛽=1), indicating a predominately diffusion-based process. Conversely, under higher biases (sub-threshold regime), the β parameter varies from 1 to 2 with ionic strength exhibiting both diffusion and drift characteristics, with a “cut point” at approximately 10−5× PBS (𝑁𝑖≈9×1014/mL). Under a high bias, once the PBS concentration reaches 10−3×, the behavior is then drift dominant. This indicates that the higher bias likely triggers electrochemical reactions and by extension, faradaic effects at most physiologically relevant ionic strengths. The signal-to-noise ratio (SNR) of the device has an inverse linear relationship with the low frequency current noise. The device exhibits a higher SNR in the sub-threshold regime than in the saturation regime. Specifically, within the saturation regime, an inversely proportional relationship between the SNR and the ionic concentration is observed. The electrochemical noise induced from ionic activities is roughly proportional to 𝑁−1/2𝑖. 

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5. Neural Network-based Channel Estimation for 2x2 and 4x4 MIMO Communication in Noisy Channels

   Yang, Kanghyeok; Vuran, Mehmet C.; Scott, Stephen; Guo, Fujuan; Ahn, Changbum R. (May 2018, International Balkan Conference on Communications and Networking)

   With increasing needs of fast and reliable commu- nication between devices, wireless communication techniques are rapidly evolving to meet such needs. Multiple input and output (MIMO) systems are one of the key techniques that utilize multiple antennas for high-throughput and reliable communication. However, increasing the number of antennas in communication also adds to the complexity of channel esti- mation, which is essential to accurately decode the transmitted data. Therefore, development of accurate and efficient channel estimation methods is necessary. We report the performance of machine learning-based channel estimation approaches to enhance channel estimation performance in high-noise envi- ronments. More specifically, bit error rate (BER) performance of 2 × 2 and 4 × 4 MIMO communication systems with space- time block coding model (STBC) and two neural network-based channel estimation algorithms is analyzed. Most significantly, the results demonstrate that a generalized regression neural network (GRNN) model matches BER results of a known-channel communication for 4 × 4 MIMO with 8-bit pilots, when trained in a specific signal to noise ratio (SNR) regime. Moreover, up to 9dB improvement in signal-to-noise ratio (SNR) for a target BER is observed, compared to least square (LS) channel estimation, especially when the model is trained in the low SNR regime. A deep artificial neural network (Deep ANN) model shows worse BER performance compared to LS in all tested environments. These preliminary results present an opportunity for achieving better performance in channel estimation through GRNN and highlight further research topics for deployment in the wild. 

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