The construction of polar codes with code length $n=2^m$ involves $m$ layers of polar transforms. In this paper, we observe that after each layer of polar transforms, one can swap certain pairs of adjacent bits to accelerate the polarization process. More precisely, if the previous bit is more reliable than its next bit under the successive decoder, then switching the decoding order of these two adjacent bits will make the reliable bit even more reliable and the noisy bit even noisier. Based on this observation, we propose a new family of codes called the Adjacent-Bits-Swapped (ABS) polar codes. We add a permutation layer after each polar transform layer in the construction of the ABS polar codes. In order to choose which pairs of adjacent bits to swap in the permutation layers, we rely on a new polar transform that combines two independent channels with $4$-ary inputs. This new polar transform allows us to track the evolution of every pair of adjacent bits through different layers of polar transforms, and it also plays an essential role in the Successive Cancellation List (SCL) decoder for the ABS polar codes. Extensive simulation results show that ABS polar codes consistently outperform standard polar codes by 0.15dB -- 0.6dB when we use CRC-aided SCL decoder with list size $32$ for both codes. The implementations of all the algorithms in this paper are available at //github.com/PlumJelly/ABS-Polar
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We present a new data-driven approach with physics-based priors to scene-level normal estimation from a single polarization image. Existing shape from polarization (SfP) works mainly focus on estimating the normal of a single object rather than complex scenes in the wild. A key barrier to high-quality scene-level SfP is the lack of real-world SfP data in complex scenes. Hence, we contribute the first real-world scene-level SfP dataset with paired input polarization images and ground-truth normal maps. Then we propose a learning-based framework with a multi-head self-attention module and viewing encoding, which is designed to handle increasing polarization ambiguities caused by complex materials and non-orthographic projection in scene-level SfP. Our trained model can be generalized to far-field outdoor scenes as the relationship between polarized light and surface normals is not affected by distance. Experimental results demonstrate that our approach significantly outperforms existing SfP models on two datasets. Our dataset and source code will be publicly available at //github.com/ChenyangLEI/sfp-wild
We present a sheaf-theoretic construction of shape space -- the space of all shapes. We do this by describing a homotopy sheaf on the poset category of constructible sets, where each set is mapped to its Persistent Homology Transform (PHT). Recent results that build on fundamental work of Schapira have shown that this transform is injective, thus making the PHT a good summary object for each shape. Our homotopy sheaf result allows us to "glue" PHTs of different shapes together to build up the PHT of a larger shape. In the case where our shape is a polyhedron we prove a generalized nerve lemma for the PHT. Finally, by re-examining the sampling result of Smale-Niyogi-Weinberger, we show that we can reliably approximate the PHT of a manifold by a polyhedron up to arbitrary precision.
Molecular communication has a key role to play in future medical applications, including detecting, analyzing, and addressing infectious disease outbreaks. Overcoming inter-symbol interference (ISI) is one of the key challenges in the design of molecular communication systems. In this paper, we propose to optimize the detection interval to minimize the impact of ISI while ensuring the accurate detection of the transmitted information symbol, which is suitable for the absorbing and passive receivers. For tractability, based on the signal-to-interference difference (SID) and signal-to-interference-and-noise amplitude ratio (SINAR), we propose a modified-SINAR (mSINAR) to measure the bit error rate (BER) performance for the molecular communication system with a variable detection interval. Besides, we derive the optimal detection interval in closed form. Using simulation results, we show that the BER performance of our proposed mSINAR scheme is superior to the competing schemes, and achieves similar performance to optimal intervals found by the exhaustive search.
Molecular mechanics (MM) potentials have long been a workhorse of computational chemistry. Leveraging accuracy and speed, these functional forms find use in a wide variety of applications in biomolecular modeling and drug discovery, from rapid virtual screening to detailed free energy calculations. Traditionally, MM potentials have relied on human-curated, inflexible, and poorly extensible discrete chemical perception rules or applying parameters to small molecules or biopolymers, making it difficult to optimize both types and parameters to fit quantum chemical or physical property data. Here, we propose an alternative approach that uses graph neural networks to perceive chemical environments, producing continuous atom embeddings from which valence and nonbonded parameters can be predicted using invariance-preserving layers. Since all stages are built from smooth neural functions, the entire process is modular and end-to-end differentiable with respect to model parameters, allowing new force fields to be easily constructed, extended, and applied to arbitrary molecules. We show that this approach is not only sufficiently expressive to reproduce legacy atom types, but that it can learn to accurately reproduce and extend existing molecular mechanics force fields. Trained with arbitrary loss functions, it can construct entirely new force fields self-consistently applicable to both biopolymers and small molecules directly from quantum chemical calculations, with superior fidelity than traditional atom or parameter typing schemes. When trained on the same quantum chemical small molecule dataset used to parameterize the openff-1.2.0 small molecule force field augmented with a peptide dataset, the resulting espaloma model shows superior accuracy vis-\`a-vis experiments in computing relative alchemical free energy calculations for a popular benchmark set.
Symbol-pair codes introduced by Cassuto and Blaum in 2010 are designed to protect against the pair errors in symbol-pair read channels. One of the central themes in symbol-error correction is the construction of maximal distance separable (MDS) symbol-pair codes that possess the largest possible pair-error correcting performance. In this paper, we construct more general generator polynomials for two classes of MDS symbol-pair codes with code length $lp$. Based on repeated-root cyclic codes, we derive all MDS symbol-pair codes of length $3p$, when the degree of the generator polynomials is no more than 10. We also give two new classes of (almost maximal distance separable) AMDS symbol-pair codes with the length $lp$ or $4p$ by virtue of repeated-root cyclic codes. For length $3p$, we derive all AMDS symbol-pair codes, when the degree of the generator polynomials is less than 10. The main results are obtained by determining the solutions of certain equations over finite fields.
Existing inferential methods for small area data involve a trade-off between maintaining area-level frequentist coverage rates and improving inferential precision via the incorporation of indirect information. In this article, we propose a method to obtain an area-level prediction region for a future observation which mitigates this trade-off. The proposed method takes a conformal prediction approach in which the conformity measure is the posterior predictive density of a working model that incorporates indirect information. The resulting prediction region has guaranteed frequentist coverage regardless of the working model, and, if the working model assumptions are accurate, the region has minimum expected volume compared to other regions with the same coverage rate. When constructed under a normal working model, we prove such a prediction region is an interval and construct an efficient algorithm to obtain the exact interval. We illustrate the performance of our method through simulation studies and an application to EPA radon survey data.
In this work, we focus on the high-dimensional trace regression model with a low-rank coefficient matrix. We establish a nearly optimal in-sample prediction risk bound for the rank-constrained least-squares estimator under no assumptions on the design matrix. Lying at the heart of the proof is a covering number bound for the family of projection operators corresponding to the subspaces spanned by the design. By leveraging this complexity result, we perform a power analysis for a permutation test on the existence of a low-rank signal under the high-dimensional trace regression model. We show that the permutation test based on the rank-constrained least-squares estimator achieves non-trivial power with no assumptions on the minimum (restricted) eigenvalue of the covariance matrix of the design. Finally, we use alternating minimization to approximately solve the rank-constrained least-squares problem to evaluate its empirical in-sample prediction risk and power of the resulting permutation test in our numerical study.
This extensive revision of my paper "Description of an $O(\text{poly}(n))$ Algorithm for NP-Complete Combinatorial Problems" will dramatically simplify the content of the original paper by solving subset-sum instead of $3$-SAT. I will first define the "product-derivative" method which will be used to generate a system of equations for solving unknown polynomial coefficients. Then I will describe the "Dragonfly" algorithm usable to solve subset-sum in $O(n^{16}\log(n))$ which is itself composed of a set of symbolic algebra steps on monic polynomials to convert a subset, $S_T$, of a set of positive integers, $S$, with a given target sum, $T$ into a polynomial with roots corresponding to the elements of $S_T$.
Most existing works of polar codes focus on the analysis of block error probability. However, in many scenarios, bit error probability is also important for evaluating the performance of channel codes. In this paper, we establish a new framework to analyze the bit error probability of polar codes. Specifically, by revisiting the error event of bit-channel, we first introduce the conditional bit error probability as a metric to evaluate the reliability of bit-channel for both systematic and non-systematic polar codes. Guided by the concept of polar subcode, we then derive an upper bound on the conditional bit error probability of each bit-channel, and accordingly, an upper bound on the bit error probability of polar codes. Based on these, two types of construction metrics aiming at minimizing the bit error probability of polar codes are proposed, which are of linear computational complexity and explicit forms. Simulation results show that the polar codes constructed by the proposed methods can outperform those constructed by the conventional methods.
Present-day atomistic simulations generate long trajectories of ever more complex systems. Analyzing these data, discovering metastable states, and uncovering their nature is becoming increasingly challenging. In this paper, we first use the variational approach to conformation dynamics to discover the slowest dynamical modes of the simulations. This allows the different metastable states of the system to be located and organized hierarchically. The physical descriptors that characterize metastable states are discovered by means of a machine learning method. We show in the cases of two proteins, Chignolin and Bovine Pancreatic Trypsin Inhibitor, how such analysis can be effortlessly performed in a matter of seconds. Another strength of our approach is that it can be applied to the analysis of both unbiased and biased simulations.
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