We systematize the approach to the investigation of deep neural network landscapes by basing it on the geometry of the space of implemented functions rather than the space of parameters. Grouping classifiers into equivalence classes, we develop a standardized parameterization in which all symmetries are removed, resulting in a toroidal topology. On this space, we explore the error landscape rather than the loss. This lets us derive a meaningful notion of the flatness of minimizers and of the geodesic paths connecting them. Using different optimization algorithms that sample minimizers with different flatness we study the mode connectivity and relative distances. Testing a variety of state-of-the-art architectures and benchmark datasets, we confirm the correlation between flatness and generalization performance; we further show that in function space flatter minima are closer to each other and that the barriers along the geodesics connecting them are small. We also find that minimizers found by variants of gradient descent can be connected by zero-error paths composed of two straight lines in parameter space, i.e. polygonal chains with a single bend. We observe similar qualitative results in neural networks with binary weights and activations, providing one of the first results concerning the connectivity in this setting. Our results hinge on symmetry removal, and are in remarkable agreement with the rich phenomenology described by some recent analytical studies performed on simple shallow models.

We demonstrate the effectiveness of frequency selective surface filters in wireless communications at low terahertz (THz) frequencies. Full-wave simulations of pass-band filters designed at 270 GHz and 330 GHz are compared with measurements over 220–360 GHz, showing remarkable agreement. The filter spectral response is used to analytically model a THz filter-based wireless channel for modulated signals. In particular, numerical results and measurements for an OOK modulated signal are in good agreement for both free-space and filtered transmission at 14 Gb/s. In both cases, bit error rates (BER) as low as 10-10 are measured. This result demonstrates that the filters marginally affect the BER with respect to free-space, interference-free transmission, whereas interfering signals are strongly rejected. This result is demonstrated through a systematic evaluation of the BER in presence of an interfering signal with different carriers and amplitudes. Results confirm a strong filter rejection to interference carriers close to the filter central frequency. Conversely, without the filters the BER performance is fully compromised. Finally, we demonstrate numerically and experimentally that the constellation diagram for 104 Gb/s QAM-16 communication is not significantly affected by the filter. The investigated filters may provide a robust approach towards efficient spectrum management for future 6G wireless applications.