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We present a unification and generalization of what is known in the literature as sequentially and hierarchically semi-separable (SSS and HSS) representations for matrices. Describing rank-structured representations of (inverses of) sparse matrices whose adjacency graph is a tree, it is shown that these so-called tree quasi-separable (TQS) matrices inherit all the favorable algebraic properties of SSS and HSS under addition, products, and inversion. To arrive at these properties, we prove a key result that characterizes the conversion of any dense matrix into a TQS representation. Here, we specifically show through an explicit construction procedure that the generator sizes are dictated by the ranks of certain Hankel blocks of the matrix. Analogous to SSS and HSS, TQS matrices admit fast matrix-vector products and direct solvers provided the generator sizes are small. A sketch of the associated algorithms is provided., Comment: 20 pages

Content creation and image editing can benefit from flexible user controls. A common intermediate representation for conditional image generation is a semantic map, that has information of objects present in the image. When compared to raw RGB pixels, the modification of semantic map is much easier. One can take a semantic map and easily modify the map to selectively insert, remove, or replace objects in the map. The method proposed in this paper takes in the modified semantic map and alter the original image in accordance to the modified map. The method leverages traditional pre-trained image-to-image translation GANs, such as CycleGAN or Pix2Pix GAN, that are fine-tuned on a limited dataset of reference images associated with the semantic maps. We discuss the qualitative and quantitative performance of our technique to illustrate its capacity and possible applications in the fields of image forgery and image editing. We also demonstrate the effectiveness of the proposed image forgery technique in thwarting the numerous deep learning-based image forensic techniques, highlighting the urgent need to develop robust and generalizable image forensic tools in the fight against the spread of fake media.

The number of malware is constantly on the rise. Though most new malware are modifications of existing ones, their sheer number is quite overwhelming. In this paper, we present a novel system to visualize and map millions of malware to points in a 2-dimensional (2D) spatial grid. This enables visualizing relationships within large malware datasets that can be used to develop triage solutions to screen different malware rapidly and provide situational awareness. Our approach links two visualizations within an interactive display. Our first view is a spatial point-based visualization of similarity among the samples based on a reduced dimensional projection of binary feature representations of malware. Our second spatial grid-based view provides a better insight into similarities and differences between selected malware samples in terms of the binary-based visual representations they share. We also provide a case study where the effect of packing on the malware data is correlated with the complexity of the packing algorithm., Comment: Submitted version - MILCOM 2022 IEEE Military Communications Conference. The high-quality images in this paper can be found on Github (https://github.com/Mayachitra-Inc/MalGrid)

Despite the success of convolutional neural networks (CNNs) in numerous computer vision tasks and their extraordinary generalization performances, several attempts to predict the generalization errors of CNNs have only been limited to a posteriori analyses thus far. A priori theories explaining the generalization performances of deep neural networks have mostly ignored the convolutionality aspect and do not specify why CNNs are able to seemingly overcome curse of dimensionality on computer vision tasks like image classification where the image dimensions are in thousands. Our work attempts to explain the generalization performance of CNNs on image classification under the hypothesis that CNNs operate on the domain of image patches. Ours is the first work we are aware of to derive an a priori error bound for the generalization error of CNNs and we present both quantitative and qualitative evidences in the support of our theory. Our patch-based theory also offers explanation for why data augmentation techniques like Cutout, CutMix and random cropping are effective in improving the generalization error of CNNs., Comment: Reorganization

Malicious PDF documents present a serious threat to various security organizations that require modern threat intelligence platforms to effectively analyze and characterize the identity and behavior of PDF malware. State-of-the-art approaches use machine learning (ML) to learn features that characterize PDF malware. However, ML models are often susceptible to evasion attacks, in which an adversary obfuscates the malware code to avoid being detected by an Antivirus. In this paper, we derive a simple yet effective holistic approach to PDF malware detection that leverages signal and statistical analysis of malware binaries. This includes combining orthogonal feature space models from various static and dynamic malware detection methods to enable generalized robustness when faced with code obfuscations. Using a dataset of nearly 30,000 PDF files containing both malware and benign samples, we show that our holistic approach maintains a high detection rate (99.92%) of PDF malware and even detects new malicious files created by simple methods that remove the obfuscation conducted by malware authors to hide their malware, which are undetected by most antiviruses., Comment: Submitted version - MILCOM 2021 IEEE Military Communications Conference

With the growing number of malware and cyber attacks, there is a need for "orthogonal" cyber defense approaches, which are complementary to existing methods by detecting unique malware samples that are not predicted by other methods. In this paper, we propose a novel and orthogonal malware detection (OMD) approach to identify malware using a combination of audio descriptors, image similarity descriptors and other static/statistical features. First, we show how audio descriptors are effective in classifying malware families when the malware binaries are represented as audio signals. Then, we show that the predictions made on the audio descriptors are orthogonal to the predictions made on image similarity descriptors and other static features. Further, we develop a framework for error analysis and a metric to quantify how orthogonal a new feature set (or type) is with respect to other feature sets. This allows us to add new features and detection methods to our overall framework. Experimental results on malware datasets show that our approach provides a robust framework for orthogonal malware detection., Comment: Submitted version - MILCOM 2021 IEEE Military Communications Conference

Seam carving is a method to resize an image in a content aware fashion. However, this method can also be used to carve out objects from images. In this paper, we propose a two-step method to detect and localize seam carved images. First, we build a detector to detect small patches in an image that has been seam carved. Next, we compute a heatmap on an image based on the patch detector's output. Using these heatmaps, we build another detector to detect if a whole image is seam carved or not. Our experimental results show that our approach is effective in detecting and localizing seam carved images.

Seam carving is a popular technique for content aware image retargeting. It can be used to deliberately manipulate images, for example, change the GPS locations of a building or insert/remove roads in a satellite image. This paper proposes a novel approach for detecting and localizing seams in such images. While there are methods to detect seam carving based manipulations, this is the first time that robust localization and detection of seam carving forgery is made possible. We also propose a seam localization score (SLS) metric to evaluate the effectiveness of localization. The proposed method is evaluated extensively on a large collection of images from different sources, demonstrating a high level of detection and localization performance across these datasets. The datasets curated during this work will be released to the public.

We consider the multi-objective optimization problem of choosing the bottom left block-entry of a block lower triangular matrix to minimize the ranks of all block sub-matrices. We provide a proof that there exists a simultaneous rank-minimizer by constructing the complete set of all minimizers., Comment: 11 pages, 1 figure, accepted in Linear Algebra and its Applications

Digital image forensics aims to detect images that have been digitally manipulated. Realistic image forgeries involve a combination of splicing, resampling, region removal, smoothing and other manipulation methods. While most detection methods in literature focus on detecting a particular type of manipulation, it is challenging to identify doctored images that involve a host of manipulations. In this paper, we propose a novel approach to holistically detect tampered images using a combination of pixel co-occurrence matrices and deep learning. We extract horizontal and vertical co-occurrence matrices on three color channels in the pixel domain and train a model using a deep convolutional neural network (CNN) framework. Our method is agnostic to the type of manipulation and classifies an image as tampered or untampered. We train and validate our model on a dataset of more than 86,000 images. Experimental results show that our approach is promising and achieves more than 0.99 area under the curve (AUC) evaluation metric on the training and validation subsets. Further, our approach also generalizes well and achieves around 0.81 AUC on an unseen test dataset comprising more than 19,740 images released as part of the Media Forensics Challenge (MFC) 2020. Our score was highest among all other teams that participated in the challenge, at the time of announcement of the challenge results.

Mixup is a procedure for data augmentation that trains networks to make smoothly interpolated predictions between datapoints. Adversarial training is a strong form of data augmentation that optimizes for worst-case predictions in a compact space around each data-point, resulting in neural networks that make much more robust predictions. In this paper, we bring these ideas together by adversarially probing the space between datapoints, using projected gradient descent (PGD). The fundamental approach in this work is to leverage backpropagation through the mixup interpolation during training to optimize for places where the network makes unsmooth and incongruous predictions. Additionally, we also explore several modifications and nuances, like optimization of the mixup ratio and geometrical label assignment, and discuss their impact on enhancing network robustness. Through these ideas, we have been able to train networks that robustly generalize better; experiments on CIFAR-10 and CIFAR-100 demonstrate consistent improvements in accuracy against strong adversaries, including the recent strong ensemble attack AutoAttack. Our source code would be released for reproducibility.

Machine Learning (ML) algorithms are susceptible to adversarial attacks and deception both during training and deployment. Automatic reverse engineering of the toolchains behind these adversarial machine learning attacks will aid in recovering the tools and processes used in these attacks. In this paper, we present two techniques that support automated identification and attribution of adversarial ML attack toolchains using Co-occurrence Pixel statistics and Laplacian Residuals. Our experiments show that the proposed techniques can identify parameters used to generate adversarial samples. To the best of our knowledge, this is the first approach to attribute gradient based adversarial attacks and estimate their parameters. Source code and data is available at: https://github.com/michael-goebel/ei_red

We propose a novel method to detect and visualize malware through image classification. The executable binaries are represented as grayscale images obtained from the count of N-grams (N=2) of bytes in the Discrete Cosine Transform (DCT) domain and a neural network is trained for malware detection. A shallow neural network is trained for classification, and its accuracy is compared with deep-network architectures such as ResNet that are trained using transfer learning. Neither dis-assembly nor behavioral analysis of malware is required for these methods. Motivated by the visual similarity of these images for different malware families, we compare our deep neural network models with standard image features like GIST descriptors to evaluate the performance. A joint feature measure is proposed to combine different features using error analysis to get an accurate ensemble model for improved classification performance. A new dataset called MaleX which contains around 1 million malware and benign Windows executable samples is created for large-scale malware detection and classification experiments. Experimental results are quite promising with 96% binary classification accuracy on MaleX. The proposed model is also able to generalize well on larger unseen malware samples and the results compare favorably with state-of-the-art static analysis-based malware detection algorithms., Comment: Submitted version - Proceedings of the 54th Hawaii International Conference on System Sciences (HICSS) 2021

We study generalization in deep learning by appealing to complexity measures originally developed in approximation and information theory. While these concepts are challenged by the high-dimensional and data-defined nature of deep learning, we show that simple vector quantization approaches such as PCA, GMMs, and SVMs capture their spirit when applied layer-wise to deep extracted features giving rise to relatively inexpensive complexity measures that correlate well with generalization performance. We discuss our results in 2020 NeurIPS PGDL challenge., Comment: Added preprint footnote

Several works in computer vision have demonstrated the effectiveness of active learning for adapting the recognition model when new unlabeled data becomes available. Most of these works consider that labels obtained from the annotator are correct. However, in a practical scenario, as the quality of the labels depends on the annotator, some of the labels might be wrong, which results in degraded recognition performance. In this paper, we address the problems of i) how a system can identify which of the queried labels are wrong and ii) how a multi-class active learning system can be adapted to minimize the negative impact of label noise. Towards solving the problems, we propose a noisy label filtering based learning approach where the inter-relationship (context) that is quite common in natural data is utilized to detect the wrong labels. We construct a graphical representation of the unlabeled data to encode these relationships and obtain new beliefs on the graph when noisy labels are available. Comparing the new beliefs with the prior relational information, we generate a dissimilarity score to detect the incorrect labels and update the recognition model with correct labels which result in better recognition performance. This is demonstrated in three different applications: scene classification, activity classification, and document classification.

Recent advances in Generative Adversarial Networks (GANs) have led to the creation of realistic-looking digital images that pose a major challenge to their detection by humans or computers. GANs are used in a wide range of tasks, from modifying small attributes of an image (StarGAN [14]), transferring attributes between image pairs (CycleGAN [91]), as well as generating entirely new images (ProGAN [36], StyleGAN [37], SPADE/GauGAN [64]). In this paper, we propose a novel approach to detect, attribute and localize GAN generated images that combines image features with deep learning methods. For every image, co-occurrence matrices are computed on neighborhood pixels of RGB channels in different directions (horizontal, vertical and diagonal). A deep learning network is then trained on these features to detect, attribute and localize these GAN generated/manipulated images. A large scale evaluation of our approach on 5 GAN datasets comprising over 2.76 million images (ProGAN, StarGAN, CycleGAN, StyleGAN and SPADE/GauGAN) shows promising results in detecting GAN generated images.

A new framework is proposed to study rank-structured matrices arising from discretizations of 2D and 3D elliptic operators. In particular, we introduce the notion of a graph-induced rank structure (GIRS) which aims to capture the fine low rank structures which appear in sparse matrices and their inverses in terms of the adjacency graph $\mathbb{G}$. We show that the GIRS property is invariant under inversion, and hence any effective representation of the inverse of GIRS matrices would lead to effective solvers. Starting with the observation that sequentially semi-separable (SSS) matrices form a good candidate for representing GIRS matrices on the line graph, we propose two extensions of SSS matrices to arbitrary graphs: Dewilde-van der Veen (DV) representations and $\mathbb{G}$-semi-separable ($\mathbb{G}$-SS) representations. It is shown that both these representations come naturally equipped with fast solvers where the solve complexity is commensurate to fast sparse Gaussian elimination on the graph $\mathbb{G}$, and $\mathbb{G}$-SS representations have a linear time multiplication algorithm. We show the construction of these representations to be highly nontrivial by determining the minimal $\mathbb{G}$-SS representation for the cycle graph $\mathbb{G}$. To obtain a minimal representation, we solve an exotic variant of a low-rank completion problem., Comment: 58 pages, 7 figures

The advent of Generative Adversarial Networks (GANs) has brought about completely novel ways of transforming and manipulating pixels in digital images. GAN based techniques such as Image-to-Image translations, DeepFakes, and other automated methods have become increasingly popular in creating fake images. In this paper, we propose a novel approach to detect GAN generated fake images using a combination of co-occurrence matrices and deep learning. We extract co-occurrence matrices on three color channels in the pixel domain and train a model using a deep convolutional neural network (CNN) framework. Experimental results on two diverse and challenging GAN datasets comprising more than 56,000 images based on unpaired image-to-image translations (cycleGAN [1]) and facial attributes/expressions (StarGAN [2]) show that our approach is promising and achieves more than 99% classification accuracy in both datasets. Further, our approach also generalizes well and achieves good results when trained on one dataset and tested on the other.

This tutorial provides an introduction to the development of fast matrix algorithms based on the notions of displacement and various low-rank structures.

The method of using periodic approximations to compute the spectral decomposition of the Koop- man operator is generalized to the class of measure-preserving flows on compact metric spaces. It is shown that the spectral decomposition of the continuous one-parameter unitary group can be approximated from an intermediate time discretization of the flow. A sufficient condition is established between the time-discretization of the flow and the spatial discretization of the periodic approximation, so that weak convergence of spectra will occur in the limit. This condition effectively translates to the requirement that the spatial refinements must occur at a faster pace than the temporal refinements. This result is contrasted with the well-known CLF condition of finite difference schemes for advection equations. Numerical results of spectral computations are shown for some benchmark examples of volume-preserving flows.

A new design methodology for neural networks that is guided by traditional algorithm design is presented. To prove our point, we present two heuristics and demonstrate an algorithmic technique for incorporating additional weights in their signal-flow graphs. We show that with training the performance of these networks can not only exceed the performance of the initial network, but can match the performance of more-traditional neural network architectures. A key feature of our approach is that these networks are initialized with parameters that provide a known performance threshold for the architecture on a given task., Comment: submitted to Thirty-second Annual Conference on Neural Information Processing Systems (NIPS), May 2018

For the class of continuous, measure-preserving automorphisms on compact metric spaces, a procedure is proposed for constructing a sequence of finite-dimensional approximations to the associated Koopman operator on a Hilbert space. These finite-dimensional approximations are obtained from the so-called "periodic approximation" of the underlying automorphism and take the form of permutation operators. Results are established on how these discretizations approximate the Koopman operator spectrally. Specificaly, it is shown that both the spectral measure and the spectral projectors of these permutation operators converge weakly to their infinite dimensional counterparts. Based on this result, a numerical method is derived for computing the spectra of volume-preserving maps on the unit $m$-torus. The discretized Koopman operator can be constructed from solving a bipartite matching problem with $\mathcal{O}(\tilde{n}^{3m/2})$ time-complexity, where $\tilde{n}$ denotes the gridsize on each dimension. By exploiting the permutation structure of the discretized Koopman operator, it is further shown that the projections and density functions are computable in $\mathcal{O}(m \tilde{n}^{m} \log \tilde{n})$ operations using the FFT algorithm. Our method is illustrated on several classical examples of automorphisms on the torus that contain either a discrete, continuous, or a mixed spectra. In addition, the spectral properties of the Chirikov standard map are examined using our method.

Realistic image forgeries involve a combination of splicing, resampling, cloning, region removal and other methods. While resampling detection algorithms are effective in detecting splicing and resampling, copy-move detection algorithms excel in detecting cloning and region removal. In this paper, we combine these complementary approaches in a way that boosts the overall accuracy of image manipulation detection. We use the copy-move detection method as a pre-filtering step and pass those images that are classified as untampered to a deep learning based resampling detection framework. Experimental results on various datasets including the 2017 NIST Nimble Challenge Evaluation dataset comprising nearly 10,000 pristine and tampered images shows that there is a consistent increase of 8%-10% in detection rates, when copy-move algorithm is combined with different resampling detection algorithms.

Resampling is an important signature of manipulated images. In this paper, we propose two methods to detect and localize image manipulations based on a combination of resampling features and deep learning. In the first method, the Radon transform of resampling features are computed on overlapping image patches. Deep learning classifiers and a Gaussian conditional random field model are then used to create a heatmap. Tampered regions are located using a Random Walker segmentation method. In the second method, resampling features computed on overlapping image patches are passed through a Long short-term memory (LSTM) based network for classification and localization. We compare the performance of detection/localization of both these methods. Our experimental results show that both techniques are effective in detecting and localizing digital image forgeries.