Chordal graphs, higher independence and vertex decomposable complexes

Given a simple undirected graph $G$ there is a simplicial complex $\mathrm{Ind}(G)$, called the independence complex, whose faces correspond to the independent sets of $G$. This is a well studied concept because it provides a fertile ground for interactions between commutative algebra, graph theory and algebraic topology. One of the line of research pursued by many authors is to determine the graph classes for which the associated independence complex is Cohen-Macaulay. For example, it is known that when $G$ is a chordal graph the complex $\mathrm{Ind}(G)$ is in fact vertex decomposable, the strongest condition in the Cohen-Macaulay ladder. In this article we consider a generalization of independence complex. Given $r\geq 1$, a subset of the vertex set is called $r$-independent if the connected components of the induced subgraph have cardinality at most $r$. The collection of all $r$-independent subsets of $G$ form a simplicial complex called the $r$-independence complex and is denoted by $\mathrm{Ind}_r(G)$. It is known that when $G$ is a chordal graph the complex $\mathrm{Ind}_r(G)$ has the homotopy type of a wedge of spheres. It is natural to ask which of the Cohen-Macaulay conditions is satisfied by these complexes. We prove, using Woodroofe's chordal hypergraph notion, that these complexes are always shellable. Further, using the notion of vertex splittable ideals we identify certain sub-classes of chordal graphs for which the associated complex is vertex decomposable.Comment: 15 pages, 4 figures. Comments are welcom

Bayesian Inference in Nonparametric Dynamic State-Space Models

We introduce state-space models where the functionals of the observational and the evolutionary equations are unknown, and treated as random functions evolving with time. Thus, our model is nonparametric and generalizes the traditional parametric state-space models. This random function approach also frees us from the restrictive assumption that the functional forms, although time-dependent, are of fixed forms. The traditional approach of assuming known, parametric functional forms is questionable, particularly in state-space models, since the validation of the assumptions require data on both the observed time series and the latent states; however, data on the latter are not available in state-space models. We specify Gaussian processes as priors of the random functions and exploit the "look-up table approach" of \ctn{Bhattacharya07} to efficiently handle the dynamic structure of the model. We consider both univariate and multivariate situations, using the Markov chain Monte Carlo (MCMC) approach for studying the posterior distributions of interest. In the case of challenging multivariate situations we demonstrate that the newly developed Transformation-based MCMC (TMCMC) of \ctn{Dutta11} provides interesting and efficient alternatives to the usual proposal distributions. We illustrate our methods with a challenging multivariate simulated data set, where the true observational and the evolutionary equations are highly non-linear, and treated as unknown. The results we obtain are quite encouraging. Moreover, using our Gaussian process approach we analysed a real data set, which has also been analysed by \ctn{Shumway82} and \ctn{Carlin92} using the linearity assumption. Our analyses show that towards the end of the time series, the linearity assumption of the previous authors breaks down.Comment: This version contains much greater clarification of the look-up table idea and a theorem regarding this is also proven and included in the supplement. Will appear in Statistical Methodolog

A nonenzymatic reduced graphene oxide-based nanosensor for parathion

Organophosphate-based pesticides (e.g., parathion (PT)) have toxic effects on human health through their residues. Therefore, cost-effective and rapid detection strategies need to be developed to ensure the consuming food is free of any organophosphate-residue. This work proposed the fabrication of a robust, nonenzymatic electrochemical-sensing electrode modified with electrochemically reduced graphene oxide (ERGO) to detect PT residues in environmental samples (e.g., soil, water) as well as in vegetables and cereals. The ERGO sensor shows a significantly affected electrocatalytic reduction peak at -0.58 V (vs Ag/AgCl) for rapid quantifi-cation of PT due to the amplified electroactive surface area of the modified electrode. At optimized experimental conditions, square-wave voltammetric analysis exhibits higher sensitivity (50.5 mu A center dot mu M-1 center dot cm(-2)), excellent selectivity, excellent stability (approximate to 180 days), good reproducibility, and repeatability for interference-free detection of PT residues in actual samples. This electro-chemical nanosensor is suitable for point-of-care detection of PT in a wide dynamic range of 3 x 10(-11)-11 x 10(-6) M with a lower detection limit of 10.9 pM. The performance of the nanosensor was validated by adding PT to natural samples and comparing the data via absorption spectroscopy. PT detection results encourage the design of easy-to-use nanosensor-based analytical tools for rapidly monitoring other environmental samples

Complex Dynamics and Synchronization of Delayed-Feedback Nonlinear Oscillators

We describe a flexible and modular delayed-feedback nonlinear oscillator that is capable of generating a wide range of dynamical behaviours, from periodic oscillations to high-dimensional chaos. The oscillator uses electrooptic modulation and fibre-optic transmission, with feedback and filtering implemented through real-time digital-signal processing. We consider two such oscillators that are coupled to one another, and we identify the conditions under which they will synchronize. By examining the rates of divergence or convergence between two coupled oscillators, we quantify the maximum Lyapunov exponents or transverse Lyapunov exponents of the system, and we present an experimental method to determine these rates that does not require a mathematical model of the system. Finally, we demonstrate a new adaptive control method that keeps two oscillators synchronized even when the coupling between them is changing unpredictably.Comment: 24 pages, 13 figures. To appear in Phil. Trans. R. Soc. A (special theme issue to accompany 2009 International Workshop on Delayed Complex Systems

GIF: Generative Interpretable Faces

Photo-realistic visualization and animation of expressive human faces have been a long standing challenge. 3D face modeling methods provide parametric control but generates unrealistic images, on the other hand, generative 2D models like GANs (Generative Adversarial Networks) output photo-realistic face images, but lack explicit control. Recent methods gain partial control, either by attempting to disentangle different factors in an unsupervised manner, or by adding control post hoc to a pre-trained model. Unconditional GANs, however, may entangle factors that are hard to undo later. We condition our generative model on pre-defined control parameters to encourage disentanglement in the generation process. Specifically, we condition StyleGAN2 on FLAME, a generative 3D face model. While conditioning on FLAME parameters yields unsatisfactory results, we find that conditioning on rendered FLAME geometry and photometric details works well. This gives us a generative 2D face model named GIF (Generative Interpretable Faces) that offers FLAME's parametric control. Here, interpretable refers to the semantic meaning of different parameters. Given FLAME parameters for shape, pose, expressions, parameters for appearance, lighting, and an additional style vector, GIF outputs photo-realistic face images. We perform an AMT based perceptual study to quantitatively and qualitatively evaluate how well GIF follows its conditioning. The code, data, and trained model are publicly available for research purposes at http://gif.is.tue.mpg.de.Comment: International Conference on 3D Vision (3DV) 202

Exemplar-Free Continual Transformer with Convolutions

Continual Learning (CL) involves training a machine learning model in a sequential manner to learn new information while retaining previously learned tasks without the presence of previous training data. Although there has been significant interest in CL, most recent CL approaches in computer vision have focused on convolutional architectures only. However, with the recent success of vision transformers, there is a need to explore their potential for CL. Although there have been some recent CL approaches for vision transformers, they either store training instances of previous tasks or require a task identifier during test time, which can be limiting. This paper proposes a new exemplar-free approach for class/task incremental learning called ConTraCon, which does not require task-id to be explicitly present during inference and avoids the need for storing previous training instances. The proposed approach leverages the transformer architecture and involves re-weighting the key, query, and value weights of the multi-head self-attention layers of a transformer trained on a similar task. The re-weighting is done using convolution, which enables the approach to maintain low parameter requirements per task. Additionally, an image augmentation-based entropic task identification approach is used to predict tasks without requiring task-ids during inference. Experiments on four benchmark datasets demonstrate that the proposed approach outperforms several competitive approaches while requiring fewer parameters.Comment: Accepted in ICCV 202