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Register a new userRandom Paraunitary Projections
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Added by
Ricardo De Queiroz
Publication date
2021
fields
Electronic Engineering
and research's language is
English
Authors
Ricardo L. de Queiroz
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Transforms using random matrices have been found to have many applications. We are concerned with the projection of a signal onto Gaussian-distributed random orthogonal bases. We also would like to easily invert the process through transposes in order to facilitate iterative reconstruction. We derive an efficient method to implement random unitary matrices of larger sizes through a set of Givens rotations. Random angles are hierarchically generated on-the-fly and the inverse merely requires traversing the angles in reverse order. Hierarchical randomization of angles also enables reduced storage. Using the random unitary matrices as building blocks we introduce random paraunitary systems (filter banks). We also highlight an efficient implementation of the paraunitary system and of its inverse. We also derive an adaptive under-decimated system, wherein one can control and adapt the amount of projections the signal undergoes, in effect, varying the sampling compression ratio as we go along the signal, without segmenting it. It may locally range from very compressive sampling matrices to (para) unitary random ones. One idea is to adapt to local sparseness characteristics of non-stationary signals.
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We introduce a novel random projection technique for efficiently reducing the dimension of very high-dimensional tensors. Building upon classical results on Gaussian random projections and Johnson-Lindenstrauss transforms~(JLT), we propose two tensorized random projection maps relying on the tensor train~(TT) and CP decomposition format, respectively. The two maps offer very low memory requirements and can be applied efficiently when the inputs are low rank tensors given in the CP or TT format. Our theoretical analysis shows that the dense Gaussian matrix in JLT can be replaced by a low-rank tensor implicitly represented in compressed form with random factors, while still approximately preserving the Euclidean distance of the projected inputs. In addition, our results reveal that the TT format is substantially superior to CP in terms of the size of the random projection needed to achieve the same distortion ratio. Experiments on synthetic data validate our theoretical analysis and demonstrate the superiority of the TT decomposition.
Let $X$ be a $d$-dimensional random vector and $X_theta$ its projection onto the span of a set of orthonormal vectors ${theta_1,...,theta_k}$. Conditions on the distribution of $X$ are given such that if $theta$ is chosen according to Haar measure on the Stiefel manifold, the bounded-Lipschitz distance from $X_theta$ to a Gaussian distribution is concentrated at its expectation; furthermore, an explicit bound is given for the expected distance, in terms of $d$, $k$, and the distribution of $X$, allowing consideration not just of fixed $k$ but of $k$ growing with $d$. The results are applied in the setting of projection pursuit, showing that most $k$-dimensional projections of $n$ data points in $R^d$ are close to Gaussian, when $n$ and $d$ are large and $k=csqrt{log(d)}$ for a small constant $c$.
Whilst adversarial attack detection has received considerable attention, it remains a fundamentally challenging problem from two perspectives. First, while threat models can be well-defined, attacker strategies may still vary widely within those constraints. Therefore, detection should be considered as an open-set problem, standing in contrast to most current detection strategies. These methods take a closed-set view and train binary detectors, thus biasing detection toward attacks seen during detector training. Second, information is limited at test time and confounded by nuisance factors including the label and underlying content of the image. Many of the current high-performing techniques use training sets for dealing with some of these issues, but can be limited by the overall size and diversity of those sets during the detection step. We address these challenges via a novel strategy based on random subspace analysis. We present a technique that makes use of special properties of random projections, whereby we can characterize the behavior of clean and adversarial examples across a diverse set of subspaces. We then leverage the self-consistency (or inconsistency) of model activations to discern clean from adversarial examples. Performance evaluation demonstrates that our technique outperforms ($>0.92$ AUC) competing state of the art (SOTA) attack strategies, while remaining truly agnostic to the attack method itself. It also requires significantly less training data, composed only of clean examples, when compared to competing SOTA methods, which achieve only chance performance, when evaluated in a more rigorous testing scenario.
We propose two training techniques for improving the robustness of Neural Networks to adversarial attacks, i.e. manipulations of the inputs that are maliciously crafted to fool networks into incorrect predictions. Both methods are independent of the chosen attack and leverage random projections of the original inputs, with the purpose of exploiting both dimensionality reduction and some characteristic geometrical properties of adversarial perturbations. The first technique is called RP-Ensemble and consists of an ensemble of networks trained on multiple project
Random projection is often used to project higher-dimensional vectors onto a lower-dimensional space, while approximately preserving their pairwise distances. It has emerged as a powerful tool in various data processing tasks and has attracted considerable research interest. Partly motivated by the recent discoveries in neuroscience, in this paper we study the problem of random projection using binary matrices with controllable sparsity patterns. Specifically, we proposed two sparse binary projection models that work on general data vectors. Compared with the conventional random projection models with dense projection matrices, our proposed models enjoy significant computational advantages due to their sparsity structure, as well as improved accuracies in empirical evaluations.
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