Kernel-based learning methods can dramatically increase the storage capacity of Hopfield networks, yet the dynamical mechanisms behind this enhancement remain poorly understood. We address this gap by combining a geometric characterization of the attractor landscape with the spectral theory of kernel machines. Using a novel metric, Pinnacle Sharpness, we empirically uncover a rich phase diagram of attractor stability, identifying a Ridge of Optimization where the network achieves maximal robustness under high-load conditions. Phenomenologically, this ridge is characterized by a Force Antagonism, in which a strong driving force is counterbalanced by a collective feedback force. We theoretically interpret this behavior as a consequence of a specific reorganization of the weight spectrum, which we term Spectral Concentration. Unlike a simple rank-1 collapse, our analysis shows that the network on the ridge self-organizes into a critical regime: the leading eigenvalue is amplified to enhance global stability (Direct Force), while the trailing eigenvalues remain finite to sustain high memory capacity (Indirect Force). Together, these results suggest a spectral mechanism by which learning reconciles stability and capacity in high-dimensional associative memory models.
Hebbian learning limits Hopfield network storage capacity (pattern-to-neuron ratio around 0.14). We propose Kernel Logistic Regression (KLR) learning. Unlike linear methods, KLR uses kernels to implicitly map patterns to high-dimensional feature space, enhancing separability. By learning dual variables, KLR dramatically improves storage capacity, achieving perfect recall even when pattern numbers exceed neuron numbers (up to ratio 1.5 shown), and enhances noise robustness. KLR demonstrably outperforms Hebbian and linear logistic regression approaches.
High-capacity associative memories based on Kernel Logistic Regression (KLR) are known for their exceptional performance but are hindered by high computational costs. This paper investigates the compressibility of KLR-trained Hopfield networks to understand the geometric principles of its robust encoding. We provide a comprehensive geometric theory based on spontaneous symmetry breaking and Walsh analysis, and validate it with compression experiments (quantization and pruning). Our experiments reveal a striking contrast: the network is extremely robust to low-precision quantization but highly sensitive to pruning. Our theory explains this via a “sparse function, dense representation” principle, where a sparse input mapping is implemented with a dense, bimodal parameterization. Our findings not only provide a practical path to hardware-efficient kernel memories but also offer new insights into the geometric principles of robust representation in neural systems.
Solar air heaters are equipment that is utilized in a variety of applications including engineering and agriculture. Improving the transfer performance of heat exchangers is necessary to benefit from efficient energy consumption. The purpose of this work is to examine the effect of novel design V-tapered-baffles on the thermal performance of solar air heaters. The experiment was conducted with the expectation of a constant wall heat flux. In turbulent flow, air serves as the testing fluid with a Reynolds number range of 6,000 to 24,000. Two V-tapered-baffle types were employed in the experiment: convergent (C-VB) and divergent (D-VB) direction V-baffles, which were employed with fixed baffle pitch length of 60 mm and four different convergent and divergent edge baffle heights of 0, 3, 6, and 9 mm. The experimental results are compared with traditional V-baffles (T-VB) and the smooth surface channel. The investigation discovered that installing V-baffles provided a better thermal performance factor than traditional V-baffles. The important factor is that the V-baffles can reduce the friction factor by about 11-50 % compared to the traditional V-baffles under the same conditions. The maximum thermal enhancement factor values for the C-VB and D-VB were 2.19 and 2.13, respectively.
