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Researchers typically investigate neural network representations by examining activation outputs for one or more layers of a network. Here, we investigate the potential for ReLU activation patterns (encoded as bit vectors) to aid in understanding and interpreting the behavior of neural networks. We utilize Representational Dissimilarity Matrices (RDMs) to investigate the coherence of data within the embedding spaces of a deep neural network. From each layer of a network, we extract and utilize bit vectors to construct similarity scores between images. From these similarity scores, we build a similarity matrix for a collection of images drawn from 2 classes. We then apply Fiedler partitioning to the associated Laplacian matrix to separate the classes. Our results indicate, through bit vector representations, that the network continues to refine class detectability with the last ReLU layer achieving better than 95\% separation accuracy. Additionally, we demonstrate that bit vectors aid in adversarial image detection, again achieving over 95\% accuracy in separating adversarial and non-adversarial images using a simple classifier., Comment: accepted by the Workshop TAG in Pattern Recognition with Applications at the Computer Vision and Pattern Recognition (CVPR) 2023

Previous work has shown that a neural network with the rectified linear unit (ReLU) activation function leads to a convex polyhedral decomposition of the input space. These decompositions can be represented by a dual graph with vertices corresponding to polyhedra and edges corresponding to polyhedra sharing a facet, which is a subgraph of a Hamming graph. This paper illustrates how one can utilize the dual graph to detect and analyze adversarial attacks in the context of digital images. When an image passes through a network containing ReLU nodes, the firing or non-firing at a node can be encoded as a bit ($1$ for ReLU activation, $0$ for ReLU non-activation). The sequence of all bit activations identifies the image with a bit vector, which identifies it with a polyhedron in the decomposition and, in turn, identifies it with a vertex in the dual graph. We identify ReLU bits that are discriminators between non-adversarial and adversarial images and examine how well collections of these discriminators can ensemble vote to build an adversarial image detector. Specifically, we examine the similarities and differences of ReLU bit vectors for adversarial images, and their non-adversarial counterparts, using a pre-trained ResNet-50 architecture. While this paper focuses on adversarial digital images, ResNet-50 architecture, and the ReLU activation function, our methods extend to other network architectures, activation functions, and types of datasets., Comment: 978-1-6654-8045-1/22/\$31.00 \copyright{}2022 IEEE The 6th Workshop on Graph Techniques for Adversarial Activity Analytics (GTA 2022)

The bulk-boundary correspondence, which links a bulk topological property of a material to the existence of robust boundary states, is a hallmark of topological insulators. However, in crystalline topological materials the presence of boundary states in the insulating gap is not always necessary since they can be hidden in the bulk energy bands, obscured by boundary artifacts of non-topological origin, or, in the case of higher-order topology, they can be gapped altogether. Crucially, in such systems the interplay between symmetry-protected topology and the corresponding symmetry defects can provide a variety of bulk probes to reveal their topological nature. For example, bulk crystallographic defects, such as disclinations and dislocations, have been shown to bind fractional charges and/or robust localized bound states in insulators protected by crystalline symmetries. Recently, exotic defects of translation symmetry called partial dislocations have been proposed as a probe of higher-order topology. However, it is a herculean task to have experimental control over the generation and probing of isolated defects in solid-state systems; hence their use as a bulk probe of topology faces many challenges. Instead, here we show that partial dislocation probes of higher-order topology are ideally suited to the context of engineered materials. Indeed, we present the first observations of partial-dislocation-induced topological modes in 2D and 3D higher-order topological insulators built from circuit-based resonator arrays. While rotational defects (disclinations) have previously been shown to indicate higher-order topology, our work provides the first experimental evidence that exotic translation defects (partial dislocations) are bulk topological probes.

We demonstrate a synthetic Hall effect for light, using an acousto-optically modulated nanophotonic resonator chain. To produce this effect, we simultaneously generate the required synthetic electric field using temporal modulation, and the required synthetic magnetic field using spatial modulation of the resonator chain. We show how the combination of these synthetic fields transverse to the direction of light propagation can be used to produce non-reciprocal optical transmission, as a basis for new photonic and topological devices.

Topological crystalline insulators (TCIs) can exhibit unique, quantized electric phenomena such as fractional electric polarization and boundary-localized fractional charge. This quantized fractional charge is the generic observable for identification of TCIs that lack robust spectral features, including ones having higher-order topology. It has been predicted that fractional charges can also manifest where crystallographic defects disrupt the lattice structure of TCIs, potentially providing a bulk probe of crystalline topology. However, this capability has not yet been confirmed in experiment since measurements of charge distributions in TCIs have not been accessible until recently. Here, we experimentally demonstrate that disclination defects can robustly trap fractional charges in TCI metamaterials, and show that this trapped charge can indicate non-trivial higher-order crystalline topology even in the absence of any spectral signatures. Furthermore, we uncover a connection between the trapped charge and the existence of topological bound states localized at these defects. We test the robustness of these topological features when the protective crystalline symmetry is broken, and find that a single robust bound state can be localized at each disclination alongside the fractional charge. Our results conclusively show that disclination defects in TCIs can robustly trap fractional charges as well as topological bound states, and moreover demonstrate the primacy of fractional charge as a probe of crystalline topology.

Spectral measurements of boundary localized in-gap modes are commonly used to identify topological insulators via the bulk-boundary correspondence. This can be extended to high-order topological insulators for which the most striking feature is in-gap modes at boundaries of higher co-dimension, e.g. the corners of a 2D material. Unfortunately, this spectroscopic approach is not always viable since the energies of the topological modes are not protected and they can often overlap the bulk bands, leading to potential misidentification. Since the topology of a material is a collective product of all its eigenmodes, any conclusive indicator of topology must instead be a feature of its bulk band structure, and should not rely on specific eigen-energies. For many topological crystalline insulators the key topological feature is fractional charge density arising from the filled bulk bands, but measurements of charge distributions have not been accessible to date. In this work, we experimentally measure boundary-localized fractional charge density of two distinct 2D rotationally-symmetric metamaterials, finding 1/4 and 1/3 fractionalization. We then introduce a new topological indicator based on collective phenomenology that allows unambiguous identification of higher-order topology, even in the absence of in-gap states. Finally, we demonstrate the higher-order bulk-boundary correspondence associated with this fractional feature by using boundary deformations to spectrally isolate localized corner modes where they were previously unobservable.

The transport of energy through 1-dimensional (1D) waveguiding channels can be affected by sub-wavelength disorder, resulting in undesirable localization and backscattering phenomena. However, quantized disorder-resilient transport is observable in the edge currents of 2-dimensional (2D) topological band insulators with broken time-reversal symmetry. Topological pumps are able to reduce this higher-dimensional topological insulator phenomena to lower dimensionality by utilizing a pumping parameter (either space or time) as an artificial dimension. Here we demonstrate the first temporal topological pump that produces on-demand, robust transport of mechanical energy using a 1D magneto-mechanical metamaterial. We experimentally demonstrate that the system is uniquely resilient to defects occurring in both space and time Our findings open a new path towards exploration of higher-dimensional topological physics with time as a synthetic dimension., Comment: Total number of pages 39, 17 pages are the main text and the rest are supplementary material. There are 5 figures in the main text and 17 figures in the supplementary section

We study nonreciprocity in spatiotemporally modulated 1D resonator chains from the perspective of equivalent 2D resonator arrays with a synthetic dimension and transverse synthetic electric and magnetic fields. The synthetic fields are respectively related to temporal and spatial modulation of the resonator chain, and we show that their combination can break transmission reciprocity without additional elements. This nonreciprocal effect is analogous to the Hall effect for charged particles. We experimentally implement chains of 2 and 3 spatiotemporally modulated resonators and measure over 58 dB of isolation contrast., Comment: 8 pages, 6 figures

The modern theory of electric polarization in crystals associates the dipole moment of an insulator with a Berry phase of its electronic ground state [1, 2]. This concept constituted a breakthrough that not only solved the long-standing puzzle of how to calculate dipole moments in crystals, but also lies at the core of the theory of topological band structures in insulators and superconductors, including the quantum anomalous Hall insulator [3, 4] and the quantum spin Hall insulator [5-7], as well as quantized adiabatic pumping processes [8-10]. A recent theoretical proposal extended the Berry phase framework to account for higher electric multipole moments [11], revealing the existence of topological phases that have not previously been observed. Here we demonstrate the first member of this predicted class -a quantized quadrupole topological insulator- experimentally produced using a GHz-frequency reconfigurable microwave circuit. We confirm the non-trivial topological phase through both spectroscopic measurements, as well as with the identification of corner states that are manifested as a result of the bulk topology. We additionally test a critical prediction that these corner states are protected by the topology of the bulk, and not due to surface artifacts, by deforming the edge between the topological and trivial regimes. Our results provide conclusive evidence of a unique form of robustness which has never previously been observed., Comment: Main Text: 14 pages, 4 figures. Supplementary Information: 6 pages, 3 figures

Inducing nonreciprocal wave propagation is a fundamental challenge across a wide range of physical systems in electromagnetics, optics, and acoustics. Recent efforts to create nonreciprocal devices have departed from established magneto-optic methods and instead exploited momentum based techniques such as coherent spatiotemporal modulation of resonators and waveguides. However, to date the nonreciprocal frequency responses that such devices can achieve have been limited, mainly to either broadband or Lorentzian-shaped transfer functions. Here we show that nonreciprocal coupling between waveguides and resonator networks enables the creation of devices with customizable nonreciprocal frequency responses. We create nonreciprocal coupling through the action of synthetic phonons, which emulate propagating phonons and can scatter light between guided and resonant modes that differ in both frequency and momentum. We implement nonreciprocal coupling in microstrip circuits and experimentally demonstrate both elementary nonreciprocal functions such as isolation and gyration as well as reconfigurable, higher-order nonreciprocal filters. Our results suggest nonreciprocal coupling as platform for a broad class of customizable nonreciprocal systems, adaptable to all wave phenomena., Comment: 19 pages, 9 figures