On the Optimal Integer-Forcing Precoding: A Geometric Perspective and a Polynomial-Time Algorithm

The joint optimization of the integer matrix $\mathbf{A}$ and the power scaling matrix $\mathbf{D}$ is central to achieving the capacity-approaching performance of Integer-Forcing (IF) precoding. This problem, however, is known to be NP-hard, presenting a fundamental computational bottleneck. In this paper, we reveal that the solution space of this problem admits a intrinsic geometric structure: it can be partitioned into a finite number of conical regions, each associated with a distinct full-rank integer matrix $\mathbf{A}$. Leveraging this decomposition, we transform the NP-hard problem into a search over these regions and propose the Multi-Cone Nested Stochastic Pattern Search (MCN-SPS) algorithm. Our main theoretical result is that MCN-SPS finds a near-optimal solution with a computational complexity of $\mathcal{O}\left(K^4\log K\log_2(r_0)\right)$, which is polynomial in the number of users $K$. Numerical simulations corroborate the theoretical analysis and demonstrate the algorithm's efficacy.

Reversible Deep Neural Network Watermarking:Matching the Floating-point Weights

Static deep neural network (DNN) watermarking embeds watermarks into the weights of DNN model by irreversible methods, but this will cause permanent damage to watermarked model and can not meet the requirements of integrity authentication. For these reasons, reversible data hiding (RDH) seems more attractive for the copyright protection of DNNs. This paper proposes a novel RDH-based static DNN watermarking method by improving the non-reversible quantization index modulation (QIM). Targeting the floating-point weights of DNNs, the idea of our RDH method is to add a scaled quantization error back to the cover object. Two schemes are designed to realize the integrity protection and legitimate authentication of DNNs. Simulation results on training loss and classification accuracy justify the superior feasibility, effectiveness and adaptability of the proposed method over histogram shifting (HS).