Optimally Sparse Frames

Frames have established themselves as a means to derive redundant, yet stable decompositions of a signal for analysis or transmission, while also promoting sparse expansions. However, when the signal dimension is large, the computation of the frame measurements of a signal typically requires a large number of additions and multiplications, and this makes a frame decomposition intractable in applications with limited computing budget. To address this problem, in this paper, we focus on frames in finite-dimensional Hilbert spaces and introduce sparsity for such frames as a new paradigm. In our terminology, a sparse frame is a frame whose elements have a sparse representation in an orthonormal basis, thereby enabling low-complexity frame decompositions. To introduce a precise meaning of optimality, we take the sum of the numbers of vectors needed of this orthonormal basis when expanding each frame vector as sparsity measure. We then analyze the recently introduced algorithm Spectral Tetris for construction of unit norm tight frames and prove that the tight frames generated by this algorithm are in fact optimally sparse with respect to the standard unit vector basis. Finally, we show that even the generalization of Spectral Tetris for the construction of unit norm frames associated with a given frame operator produces optimally sparse frames

Bit-Metric Decoding of Non-Binary LDPC Codes with Probabilistic Amplitude Shaping

A new approach for combining non-binary low-density parity-check (NB-LDPC) codes with higher-order modulation and probabilistic amplitude shaping (PAS) is presented. Instead of symbol-metric decoding (SMD), a bit-metric decoder (BMD) is used so that matching the field order of the non-binary code to the constellation size is not needed, which increases the flexibility of the coding scheme. Information rates, density evolution thresholds and finite-length simulations show that the flexibility comes at no loss of performance if PAS is used.Comment: Accepted for IEEE Communication Letter

Digital Multimedia Forensics and Anti-Forensics

As the use of digital multimedia content such as images and video has increased, so has the means and the incentive to create digital forgeries. Presently, powerful editing software allows forgers to create perceptually convincing digital forgeries. Accordingly, there is a great need for techniques capable of authenticating digital multimedia content. In response to this, researchers have begun developing digital forensic techniques capable of identifying digital forgeries. These forensic techniques operate by detecting imperceptible traces left by editing operations in digital multimedia content. In this dissertation, we propose several new digital forensic techniques to detect evidence of editing in digital multimedia content. We begin by identifying the fingerprints left by pixel value mappings and show how these can be used to detect the use of contrast enhancement in images. We use these fingerprints to perform a number of additional forensic tasks such as identifying cut-and-paste forgeries, detecting the addition of noise to previously JPEG compressed images, and estimating the contrast enhancement mapping used to alter an image. Additionally, we consider the problem of multimedia security from the forger's point of view. We demonstrate that an intelligent forger can design anti-forensic operations to hide editing fingerprints and fool forensic techniques. We propose an anti-forensic technique to remove compression fingerprints from digital images and show that this technique can be used to fool several state-of-the-art forensic algorithms. We examine the problem of detecting frame deletion in digital video and develop both a technique to detect frame deletion and an anti-forensic technique to hide frame deletion fingerprints. We show that this anti-forensic operation leaves behind fingerprints of its own and propose a technique to detect the use of frame deletion anti-forensics. The ability of a forensic investigator to detect both editing and the use of anti-forensics results in a dynamic interplay between the forger and forensic investigator. We use develop a game theoretic framework to analyze this interplay and identify the set of actions that each party will rationally choose. Additionally, we show that anti-forensics can be used protect against reverse engineering. To demonstrate this, we propose an anti-forensic module that can be integrated into digital cameras to protect color interpolation methods

Learning World Models with Identifiable Factorization

Extracting a stable and compact representation of the environment is crucial for efficient reinforcement learning in high-dimensional, noisy, and non-stationary environments. Different categories of information coexist in such environments -- how to effectively extract and disentangle these information remains a challenging problem. In this paper, we propose IFactor, a general framework to model four distinct categories of latent state variables that capture various aspects of information within the RL system, based on their interactions with actions and rewards. Our analysis establishes block-wise identifiability of these latent variables, which not only provides a stable and compact representation but also discloses that all reward-relevant factors are significant for policy learning. We further present a practical approach to learning the world model with identifiable blocks, ensuring the removal of redundants but retaining minimal and sufficient information for policy optimization. Experiments in synthetic worlds demonstrate that our method accurately identifies the ground-truth latent variables, substantiating our theoretical findings. Moreover, experiments in variants of the DeepMind Control Suite and RoboDesk showcase the superior performance of our approach over baselines