Spectrally-normalized margin bounds for neural networks

This paper presents a margin-based multiclass generalization bound for neural networks that scales with their margin-normalized "spectral complexity": their Lipschitz constant, meaning the product of the spectral norms of the weight matrices, times a certain correction factor. This bound is empirically investigated for a standard AlexNet network trained with SGD on the mnist and cifar10 datasets, with both original and random labels; the bound, the Lipschitz constants, and the excess risks are all in direct correlation, suggesting both that SGD selects predictors whose complexity scales with the difficulty of the learning task, and secondly that the presented bound is sensitive to this complexity.Comment: Comparison to arXiv v1: 1-norm in main bound refined to (2,1)-group-norm. Comparison to NIPS camera ready: typo fixe

Link and subgraph likelihoods in random undirected networks with fixed and partially fixed degree sequence

The simplest null models for networks, used to distinguish significant features of a particular network from {\it a priori} expected features, are random ensembles with the degree sequence fixed by the specific network of interest. These "fixed degree sequence" (FDS) ensembles are, however, famously resistant to analytic attack. In this paper we introduce ensembles with partially-fixed degree sequences (PFDS) and compare analytic results obtained for them with Monte Carlo results for the FDS ensemble. These results include link likelihoods, subgraph likelihoods, and degree correlations. We find that local structural features in the FDS ensemble can be reasonably well estimated by simultaneously fixing only the degrees of few nodes, in addition to the total number of nodes and links. As test cases we use a food web, two protein interaction networks (\textit{E. coli, S. cerevisiae}), the internet on the autonomous system (AS) level, and the World Wide Web. Fixing just the degrees of two nodes gives the mean neighbor degree as a function of node degree, $_k$, in agreement with results explicitly obtained from rewiring. For power law degree distributions, we derive the disassortativity analytically. In the PFDS ensemble the partition function can be expanded diagrammatically. We obtain an explicit expression for the link likelihood to lowest order, which reduces in the limit of large, sparse undirected networks with $L$ links and with $k_{\rm max} \ll L$ to the simple formula $P(k,k') = kk'/(2L + kk')$. In a similar limit, the probability for three nodes to be linked into a triangle reduces to the factorized expression $P_{\Delta}(k_1,k_2,k_3) = P(k_1,k_2)P(k_1,k_3)P(k_2,k_3)$.Comment: 17 pages, includes 11 figures; first revision: shortened to 14 pages (7 figures), added discussion of subgraph counts, deleted discussion of directed network

The chloroplast land plant phylogeny: analyses employing better-fitting tree- and site-heterogeneous composition models

The colonization of land by descendants of charophyte green algae marked a turning point in Earth history that enabled the development of the diverse terrestrial ecosystems we see today. Early land plants diversified into three gametophyte-dominant lineages, namely the hornworts, liverworts, and mosses, collectively known as bryophytes, and a sporophyte-dominant lineage, the vascular plants, or tracheophytes. In recent decades, the prevailing view of evolutionary relationships among these four lineages has been that the tracheophytes were derived from a bryophyte ancestor. However, recent phylogenetic evidence has suggested that bryophytes are monophyletic, and thus that the first split among land plants gave rise to the lineages that today we recognize as the bryophytes and tracheophytes. We present a phylogenetic analysis of chloroplast protein-coding data that also supports the monophyly of bryophytes. This newly compiled data set consists of 83 chloroplast genes sampled across 30 taxa that include chlorophytes and charophytes, including four members of the Zygnematophyceae, and land plants, that were sampled following a balanced representation of the main bryophyte and tracheophyte lineages. Analyses of non-synonymous site nucleotide data and amino acid translation data result in congruent phylogenetic trees showing the monophyly of bryophytes, with the Zygnematophyceae as the charophyte group most closely related to land plants. Analyses showing that bryophytes and tracheophytes evolved separately from a common terrestrial ancestor have profound implications for the way we understand the evolution of plant life cycles on land and how we interpret the early land plant fossil record.This work was supported by FCT (Portuguese Foundation for Science and Technology) through project grant PTDC/BIA-EVF/1499/2014 to CC and national funds through project UIDB/04326/2020, and from the operational programs CRESC Algarve 2020 and COMPETE 2020 through projects EMBRC.PT ALG-01-0145-FEDER-022121 and BIODATA.PT ALG-01-0145-FEDER-022231.info:eu-repo/semantics/publishedVersio

Clustering Phase Transitions and Hysteresis: Pitfalls in Constructing Network Ensembles

Ensembles of networks are used as null models in many applications. However, simple null models often show much less clustering than their real-world counterparts. In this paper, we study a model where clustering is enhanced by means of a fugacity term as in the Strauss (or "triangle") model, but where the degree sequence is strictly preserved -- thus maintaining the quenched heterogeneity of nodes found in the original degree sequence. Similar models had been proposed previously in [R. Milo et al., Science 298, 824 (2002)]. We find that our model exhibits phase transitions as the fugacity is changed. For regular graphs (identical degrees for all nodes) with degree k > 2 we find a single first order transition. For all non-regular networks that we studied (including Erdos - Renyi and scale-free networks) we find multiple jumps resembling first order transitions, together with strong hysteresis. The latter transitions are driven by the sudden emergence of "cluster cores": groups of highly interconnected nodes with higher than average degrees. To study these cluster cores visually, we introduce q-clique adjacency plots. We find that these cluster cores constitute distinct communities which emerge spontaneously from the triangle generating process. Finally, we point out that cluster cores produce pitfalls when using the present (and similar) models as null models for strongly clustered networks, due to the very strong hysteresis which effectively leads to broken ergodicity on realistic time scales.Comment: 13 pages, 11 figure

The Structure of a Langmuir Monolayer of Methyl Eicosanoate as Determined by X‐Ray Diffraction and Brewster Angle Microscopy

Relaxed Langmuir monolayers of methyl eicosanoate were studied using Brewster angle microscopy (BAM) and grazing incidence x‐ray diffraction (GIXD). The structure of the various phases in this system was determined and compared to previous isotherm and microscopy measurements. At low pressure and low temperature, a crystalline phase with tilt toward nearest neighbor, I($L^{\prime\prime}_{2}$), is observed. At a temperature of $\sim$14$^\circ$C and low pressure there is a transition to a phase with tilt toward next‐nearest neighbor, F($L^{\prime}_{2}$). Finally, as the temperature continues to be raised at low pressure, there is a transition to a phase where the two F($L^{\prime}_{2}$) peaks have the same values of $Q_{xy}$, here called the $\tau$ phase, with a Rotator IV‐like structure. At high pressure and low temperature, an untilted, orthorhombic phase, $U^{\prime}$(CS), is observed. As the temperature is increased, a second untilted, orthorhombic phase, U(S), is observed, beginning at $\sim$12$^\circ$C. Above 21$^\circ$C and at high pressure, a Rotator‐II‐like phase is seen, with an undistorted, untilted structure. Finally, results from BAM measurements are used to monitor the texture of the film in the different phases.Engineering and Applied Science

The mitochondrial phylogeny of land plants shows support for Setaphyta under composition-heterogeneous substitution models

Congruence among analyses of plant genomic data partitions (nuclear, chloroplast and mitochondrial) is a strong indicator of accuracy in plant molecular phylogenetics. Recent analyses of both nuclear and chloroplast genome data of land plants (embryophytes) have, controversially, been shown to support monophyly of both bryophytes (mosses, liverworts, and hornworts) and tracheophytes (lycopods, ferns, and seed plants), with mosses and liverworts forming the clade Setaphyta. However, relationships inferred from mitochondria are incongruent with these results, and typically indicate paraphyly of bryophytes with liverworts alone resolved as the earliest-branching land plant group. Here, we reconstruct the mitochondrial land plant phylogeny from a newly compiled data set. When among-lineage composition heterogeneity is accounted for in analyses of codon-degenerate nucleotide and amino acid data, the clade Setaphyta is recovered with high support, and hornworts are supported as the earliest-branching lineage of land plants. These new mitochondrial analyses demonstrate partial congruence with current hypotheses based on nuclear and chloroplast genome data, and provide further incentive for revision of how plants arose on land.UIDB/04326/2020, PTDC/BIA-EVF/1499/2014, EMBRC.PT ALG-01-0145-FEDER-022121, BIODATA.PT ALG-01-0145-FEDER-022231info:eu-repo/semantics/publishedVersio

The 'double face' illusion

We report three experiments intended to characterise aspects of the ‘double’ face illusion, formed by replicating the eyes and mouth below the originals. Such doubled faces are disturbing to look at. We find there are wide individual differences in ability to detect that a face has been doubled when presented briefly and masked. These differences appear to relate to perceptual speed, since they correlate with the ability to identify a briefly presented famous face. Doubling has a significant effect on identification, though much less than inversion. In a reaction time study, participants are faster to decide that a face has been doubled as it is rotated away from upright. The final study shows that normal and doubled faces do not pop out from each other, but reveals a processing overhead of 40-60ms per doubled face. We offer some speculations as to the cause of the perceptual effects