The natural gradient method is widely used in statistical optimization, but its standard formulation assumes a Euclidean parameter space. This paper proposes an inversion-free stochastic natural gradient method for probability distributions whose parameters lie on a Riemannian manifold. The manifold setting offers several advantages: one can implicitly enforce parameter constraints such as positive definiteness and orthogonality, ensure parameters are identifiable, or guarantee regularity properties of the objective like geodesic convexity. Building on an intrinsic formulation of the Fisher information matrix (FIM) on a manifold, our method maintains an online approximation of the inverse FIM, which is efficiently updated at quadratic cost using score vectors sampled at successive iterates. In the Riemannian setting, these score vectors belong to different tangent spaces and must be combined using transport operations. We prove almost-sure convergence rates of $O(\log{s}/s^α)$ for the squared distance to the minimizer when the step size exponent $α>2/3$. We also establish almost-sure rates for the approximate FIM, which now accumulates transport-based errors. A limited-memory variant of the algorithm with sub-quadratic storage complexity is proposed. Finally, we demonstrate the effectiveness of our method relative to its Euclidean counterparts on variational Bayes with Gaussian approximations and normalizing flows.

Abstract--The 2024 July Revolution in Bangladesh represents a landmark event in the study of civil resistance: a successful, student-led civilian uprising that overthrew a long-standing authoritarian regime despite facing brutal state repression. This study investigates the central paradox of its success: how state violence, intended to quell dissent, ultimately fueled the movement's victory. We employ a mixed-methods approach. First, we develop a qualitative narrative of the conflict's timeline to generate specific, testable hypotheses. Then, using a disaggregated, event-level dataset, we employ a multi-method quantitative analysis to dissect the complex relationship between repression and mobilisation. We provide a framework to analyse explosive modern uprisings like the July Revolution. Initial pooled regression models highlight the crucial role of protest momentum (measured by a feedback loop effect) in sustaining the movement. T o isolate causal effects, we specify a Two-Way Fixed Effects panel model, which provides robust evidence for a direct and statistically significant local suppression backfire effect. Our V ector Autoregression (V AR) analysis provides clear visual evidence of an immediate, nationwide mobilisation in response to increased lethal violence. We further demonstrate that this effect was non-linear . A structural break analysis reveals that the backfire dynamic was statistically insignificant in the conflict's early phase but was triggered by the catalytic moral shock of the first wave of lethal violence, and its visuals circulated around July 16th. We conclude that the July Revolution was driven by a contingent, non-linear backfire, triggered by specific catalytic moral shocks and accelerated by the viral reaction to the visual spectacle of state brutality. N August 2024, the fifteen-year rule of Prime Minister Sheikh Hasina of Bangladesh came to a sudden and dramatic end. After weeks of escalating nationwide protests, she resigned from her post and fled the country. These authors contributed equally to this work. Saiful Bari Siddiqui is a Senior Lecturer at the Department of Computer Science and Engineering, BRAC University, Dhaka, Bangladesh (e-mail: saiful.bari@bracu.ac.bd). Anupam Debashis Roy is a PhD candidate at the Department of Sociology, University of Oxford, Oxford, United Kingdom (e-mail: anu-pam.roy@sant.ox.ac.uk). In a matter of weeks, this initial spark grew into a nationwide fire, as hundreds of thousands of ordinary citizens joined the students, bringing the country to a standstill and achieving a political transformation that had seemed unthinkable just a month earlier.

Belarus and Russia's show of firepower appears to be a message to Europe In a large field 45 miles (72km) from Belarus' capital Minsk, a battle is raging. There are giant explosions as Sukhoi-34 bombers drop guided bombs. Helicopter gunships join the attack, while surveillance drones sweep overhead to view the damage. Together with other international media we've been brought to the Borisovsky training ground where Belarusian and Russian forces are taking part in joint manoeuvres. Military attachés, too, from a variety of embassies are observing the drill from a viewing platform.

How is Russia replenishing its military? What is a'coalition of the willing'? How China forgot promises and'debts' to Ukraine How are Europe, the US pulling apart on Ukraine? Russia and Belarus have begun large-scale military exercises, raising alarm across NATO's eastern flank just days after Warsaw accused Moscow of sending attack drones across Polish airspace, a major escalation that sent shivers through Europe. The Zapad 2025 manoeuvres, which run from Friday until Tuesday, are taking place as Russian forces continue their slow advance in Ukraine and intensify air attacks on Ukrainian cities.

How is Russia replenishing its military? What is a'coalition of the willing'? How China forgot promises and'debts' to Ukraine How are Europe, the US pulling apart on Ukraine? Anti-aircraft units downed seven Ukrainian drones headed for Moscow early on Friday, according to the Russian capital's mayor Sergei Sobyanin. Russian forces have taken control of the settlement of Sosnivka in Ukraine's Dnipropetrovsk region, Russia's Defence Ministry said on Thursday.

Dozens of political prisoners have been freed from Belarusian prisons as part of a deal between authoritarian leader Alexander Lukashenko and US President Donald Trump. Fifty-two prisoners have been released, including trade union leaders, journalists and activists, but more than 1,000 political prisoners remain in jail. In exchange, the US has said it will relieve some sanctions on Belarusian airline Belavia, allowing it to buy parts for its airlines. The prisoner release came on the eve of joint military exercises involving Belarus and close ally Russia, and after what neighbouring Poland called an unprecedented Russian drone incursion into its airspace. Poland is closing its borders with Belarus because of the Zapad-2025 drills, which last until Tuesday.

Here's What to Know About Poland Shooting Down Russian Drones On Wednesday morning, Poland shot down several Russian drones that entered its airspace--a first since Moscow's invasion of Ukraine. The incident disrupted air travel and set the region on edge. Airports closed in Poland after the country's military detected Russian drones in its airspace. Early Wednesday morning, Poland shot down several Russian drones that had violated its airspace during a massive strike against western Ukraine . The Polish military operation, confirmed by Prime Minister Donald Tusk through a social media message in the early morning hours, marks a turning point in Warsaw's involvement in the conflict that has affected the region for more than two and a half years.

Ukraine said it carried out an overnight drone strike on the Kirovske airfield in Crimea and claimed that multiple Russian helicopters and an air defence system were destroyed in the strike. According to a Ukraine Security Service (SBU) statement, the drones targeted areas where Russian aviation units, air defence assets, ammunition depots and unmanned aerial vehicles were located. The agency claimed that Mi-8, Mi-26, and Mi-28 helicopters, as well as a Pantsir-S1 missile and gun system were destroyed. "Secondary detonations continued throughout the night at the airfield," the SBU said, calling the strike part of broader efforts to disrupt Russian aerial operations. "The enemy must understand that expensive military equipment and ammunition are not safe anywhere – not on the line of contact, not in Crimea, and not deep in the rear."

Understanding the complex interactions within dynamic multilayer networks is critical for advancements in various scientific domains. Existing models often fail to capture such networks' temporal and cross-layer dynamics. This paper introduces a novel Tensor State Space Model for Dynamic Multilayer Networks (TSSDMN), utilizing a latent space model framework. TSSDMN employs a symmetric Tucker decomposition to represent latent node features, their interaction patterns, and layer transitions. Then by fixing the latent features and allowing the interaction patterns to evolve over time, TSSDMN uniquely captures both the temporal dynamics within layers and across different layers. The model identifiability conditions are discussed. By treating latent features as variables whose posterior distributions are approximated using a mean-field variational inference approach, a variational Expectation Maximization algorithm is developed for efficient model inference. Numerical simulations and case studies demonstrate the efficacy of TSSDMN for understanding dynamic multilayer networks.

The paper presents a learned two-dimensional separable transform (LST) that can be considered as a new type of computational layer for constructing neural network (NN) architecture for image recognition tasks. The LST based on the idea of sharing the weights of one fullyconnected (FC) layer to process all rows of an image. After that, a second shared FC layer is used to process all columns of image representation obtained from the first layer. The use of LST layers in a NN architecture significantly reduces the number of model parameters compared to models that use stacked FC layers. We show that a NN-classifier based on a single LST layer followed by an FC layer achieves 98.02\% accuracy on the MNIST dataset, while having only 9.5k parameters. We also implemented a LST-based classifier for handwritten digit recognition on the FPGA platform to demonstrate the efficiency of the suggested approach for designing a compact and high-performance implementation of NN models. Git repository with supplementary materials: https://github.com/Mak-Sim/LST-2d