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Score matching has become a central training objective in modern generative modeling, particularly in diffusion models, where it is used to learn high-dimensional data distributions through the estimation of score functions. Despite its empirical success, the theoretical understanding of the optimization behavior of score matching, particularly in over-parameterized regimes, remains limited. In this work, we study gradient descent for training over-parameterized models to learn a single Gaussian distribution. Specifically, we use a student model with n learnable parameters, motivated by the structure of a Gaussian mixture model, and train it on data generated from a single ground-truth Gaussian using the population score matching objective. We analyze the optimization dynamics under multiple regimes. When the noise scale is sufficiently large, we prove a global convergence result for gradient descent, which resembles the known behavior of gradient EM in over-parameterized settings. In the low-noise regime, we identify the existence of a stationary point, highlighting the difficulty of proving global convergence in this case. Nevertheless, we show convergence under certain initialization conditions: when the parameters are initialized to be exponentially small, gradient descent ensures convergence of all parameters to the ground truth. We further give an example where, without the exponentially small initialization, the parameters may not converge to the ground truth. Finally, we consider the case of random initialization, where parameters are sampled from a Gaussian distribution far from the ground truth. We prove that, with high probability, only one parameter converges while the others diverge to infinity, yet the loss still converges to zero with a 1/ω rate, where ω is the number of iterations. We also establish a nearly matching lower bound on the convergence rate in this regime. This is the first work to establish global convergence guarantees for Gaussian mixtures with at least three components under the score matching framework. 

Sodium naphthenates (NaNs), found in crude oils and oil sands process-affected water (OSPW), can act as surfactants and stabilize undesirable foams and emulsions. Despite the critical impact of soap-like NaNs on the formation, properties, and stability of petroleum and OSPW foams, there is a significant lack of studies that characterize foam film drainage, motivating this study. Here, we contrast the drainage of aqueous foam films formulated with NaN with foams containing sodium dodecyl sulfate (SDS), a well-studied surfactant system, in the relatively low concentration regime ( c /CMC < 12.5). The foam films exhibit drainage via stratification, displaying step-wise thinning and coexisting thick–thin regions manifested as distinct shades of gray in reflected light microscopy due to thickness-dependent interference intensity. Using IDIOM (interferometry digital imaging optical microscopy) protocols that we developed, we analyze pixel-wise intensity to obtain thickness maps with high spatiotemporal resolution (thickness <1 nm, lateral ∼500 nm, time ∼10 ms). The analysis of interference intensity variations over time reveals that the aqueous foam films of both SDS and NaN possess an evolving, dynamic, and rich nanoscopic topography. The nanoscopic thickness transitions for stratifying SDS foam films are attributed to the role played by damped supramolecular oscillatory structural disjoining pressure contributed by the confinement-induced layering of spherical micelles. In comparison with SDS, we find smaller concentration-dependent step size and terminal film thickness values for NaN, implying weaker intermicellar interactions and oscillatory structural disjoining pressure with shorter decay length and periodicity. 

Transport measurements indicate superconductivity in twisted trilayers, quadrilayers, and pentalayers of graphene. 

We report the discovery of a hitherto unreported mechanism of drainage and rupture of micellar foam films that presents unexplored opportunities for understanding and controlling the stability, lifetime and properties of ubiquitous foams. It is well-known that ultrathin micellar foam films exhibit stratification, manifested as stepwise thinning and coexistence of thin–thick flat regions that differ in thickness by a nanoscopic step size equal to the intermicellar distance. Stratification typically involves the spontaneous formation and growth of thinner, darker, circular domains or thicker, brighter mesas. Mechanistically, domain expansion appears similar to hole growth in polymer films undergoing dewetting by nucleation and growth mechanism that can be described by considering metastable states resulting from a thickness-dependent oscillatory free energy. Dewetting polymer films occasionally phase separate into thick and thin regions forming an interconnected, network-like morphology by undergoing spinodal dewetting. However, the formation of thick–thin spinodal patterns has never been reported for freestanding films. In this contribution, we show that the thickness-dependent oscillatory contribution to free energy that arises due to confinement-induced layering of micelles can drive the formation of such thick-thin regions by undergoing a process we term as spinodal stratification. We visualize and characterize the nanoscopic thickness variations and transitions by using IDIOM (interferometry digital imaging optical microscopy) protocols to obtain exquisite thickness maps of freestanding films. We find that evaporation and enhanced drainage in vertical films play a critical role in driving the process, and spinodal stratification can occur in both single foam films and in bulk foam.