Models and information-theoretic bounds for nanopore sequencing

Nanopore sequencing is an emerging new technology for sequencing DNA, which can read long fragments of DNA (~50,000 bases) in contrast to most current short-read sequencing technologies which can only read hundreds of bases. While nanopore sequencers can acquire long reads, the high error rates (20%-30%) pose a technical challenge. In a nanopore sequencer, a DNA is migrated through a nanopore and current variations are measured. The DNA sequence is inferred from this observed current pattern using an algorithm called a base-caller. In this paper, we propose a mathematical model for the "channel" from the input DNA sequence to the observed current, and calculate bounds on the information extraction capacity of the nanopore sequencer. This model incorporates impairments like (non-linear) inter-symbol interference, deletions, as well as random response. These information bounds have two-fold application: (1) The decoding rate with a uniform input distribution can be used to calculate the average size of the plausible list of DNA sequences given an observed current trace. This bound can be used to benchmark existing base-calling algorithms, as well as serving a performance objective to design better nanopores. (2) When the nanopore sequencer is used as a reader in a DNA storage system, the storage capacity is quantified by our bounds

Learning Temporal Dependence from Time-Series Data with Latent Variables

We consider the setting where a collection of time series, modeled as random processes, evolve in a causal manner, and one is interested in learning the graph governing the relationships of these processes. A special case of wide interest and applicability is the setting where the noise is Gaussian and relationships are Markov and linear. We study this setting with two additional features: firstly, each random process has a hidden (latent) state, which we use to model the internal memory possessed by the variables (similar to hidden Markov models). Secondly, each variable can depend on its latent memory state through a random lag (rather than a fixed lag), thus modeling memory recall with differing lags at distinct times. Under this setting, we develop an estimator and prove that under a genericity assumption, the parameters of the model can be learned consistently. We also propose a practical adaption of this estimator, which demonstrates significant performance gains in both synthetic and real-world datasets

ClusterGAN : Latent Space Clustering in Generative Adversarial Networks

Generative Adversarial networks (GANs) have obtained remarkable success in many unsupervised learning tasks and unarguably, clustering is an important unsupervised learning problem. While one can potentially exploit the latent-space back-projection in GANs to cluster, we demonstrate that the cluster structure is not retained in the GAN latent space. In this paper, we propose ClusterGAN as a new mechanism for clustering using GANs. By sampling latent variables from a mixture of one-hot encoded variables and continuous latent variables, coupled with an inverse network (which projects the data to the latent space) trained jointly with a clustering specific loss, we are able to achieve clustering in the latent space. Our results show a remarkable phenomenon that GANs can preserve latent space interpolation across categories, even though the discriminator is never exposed to such vectors. We compare our results with various clustering baselines and demonstrate superior performance on both synthetic and real datasets.Comment: GANs, Clustering, Latent Space, Interpolation (v2 : Typos fixed, some new experiments added, reported metrics on best validated model.

Minimum HGR Correlation Principle: From Marginals to Joint Distribution

Given low order moment information over the random variables $\mathbf{X} = (X_1,X_2,\ldots,X_p)$ and $Y$, what distribution minimizes the Hirschfeld-Gebelein-R\'{e}nyi (HGR) maximal correlation coefficient between $\mathbf{X}$ and $Y$, while remains faithful to the given moments? The answer to this question is important especially in order to fit models over $(\mathbf{X},Y)$ with minimum dependence among the random variables $\mathbf{X}$ and $Y$. In this paper, we investigate this question first in the continuous setting by showing that the jointly Gaussian distribution achieves the minimum HGR correlation coefficient among distributions with the given first and second order moments. Then, we pose a similar question in the discrete scenario by fixing the pairwise marginals of the random variables $\mathbf{X}$ and $Y$. To answer this question in the discrete setting, we first derive a lower bound for the HGR correlation coefficient over the class of distributions with fixed pairwise marginals. Then we show that this lower bound is tight if there exists a distribution with certain {\it additive} structure satisfying the given pairwise marginals. Moreover, the distribution with the additive structure achieves the minimum HGR correlation coefficient. Finally, we conclude by showing that the event of obtaining pairwise marginals containing an additive structured distribution has a positive Lebesgue measure over the probability simplex