Abstract In visual culture, Generative Adversarial Networks (GANs) have been used to generate two-dimensional images, video, and three-dimensional forms. Whilst there are a relatively large number of conditional datasets of two-dimensional images, there are fewer datasets of three-dimensional objects available for training, evaluation, and practical use. In this paper, we introduce a synthetic dataset of 3D trees that provide novel geometric challenges for machine learning and describe our use of this dataset in training and qualitative evaluation via a public exhibition and survey. The 3D Tree Dataset was made using random variations of 76 bespoke tree templates based on art historical references to favor interesting, beautiful, and varied trunk and branch shapes. In generating and experimenting with this dataset, we wanted to know how a geometrically complex series of organic forms would challenge voxel-GANs compared to existing datasets comprising industrial objects such as chairs and cars. As an interdisciplinary project between visual arts and computer science, we used 3D printing and animation to visualize and communicate our outputs to non-specialist audiences. In this paper, we describe and share the 3D Tree Dataset, outline the process of generating the dataset, describe the GAN architecture and application used to test the dataset and discuss the results relative to our artistic and cultural goals.

A simple binary matroid is called $I_4$-free if none of its rank-4 flats are independent sets. These objects can be equivalently defined as the sets $E$ of points in $\mbox{PG}(n-1,2)$ for which $E \cap F$ is not a basis of $F$ for any four-dimensional flat $F$. We prove a decomposition theorem that exactly determines the structure of all $I_4$-free and triangle-free matroids. In particular, our theorem implies that the $I_4$-free and triangle-free matroids have critical number at most 2.

We prove that for every proper minor-closed class $\mathcal{M}$ of $\mathbb{F}_p$-representable matroids, there exists an $O(1)$-competitive algorithm for the matroid secretary problem on $\mathcal{M}$. This result relies on the extremely powerful matroid minor structure theory being developed by Geelen, Gerards, and Whittle. We also note that, for asymptotically almost all matroids, the matroid secretary algorithm that selects a random basis, ignoring weights, is $(2+o(1))$-competitive. In fact, assuming the conjecture that almost all matroids are paving, there is a $(1+o(1))$-competitive algorithm for almost all matroids.

A biased graph is a pair $(G,\mathcal{B})$, where $G$ is a graph and $\mathcal{B}$ is a collection of “balanced” cycles of $G$ such that no $\Theta$-subgraph of $G$ contains precisely two balanced cycles. We prove a Ramsey-type theorem, showing that if $(G,\mathcal{B})$ is a biased graph for which $G$ is a very large complete graph, then $G$ contains a large complete subgraph $H$ such that the set of balanced cycles within $H$ has one of three specific, highly symmetric structures, all of which can be described naturally via group-labelings.

A biased graph is a graph $G$, together with a distinguished subset $\mathcal{B}$ of its cycles, so that no theta-subgraph of $G$ contains precisely two cycles in $\mathcal{B}$. A large number of biased graphs can be constructed by choosing $G$ to be a complete graph, and $\mathcal{B}$ to be an arbitrary subset of its Hamilton cycles. We show that, on the logarithmic scale, the total number of simple biased graphs on $n$ vertices does not asymptotically exceed the number that can be constructed in this elementary way.

A matroid is Ingleton if all quadruples of subsets of its ground set satisfy Ingleton's inequality. In particular, representable matroids are Ingleton. We show that the number of Ingleton matroids on ground set $[n]$ is doubly exponential in $n$; it follows that almost all Ingleton matroids are nonrepresentable.

For a class C of binary linear codes, we write θC: (0, 1) → [0, (1/2)] for the maximum-likelihood decoding threshold function of C, the function whose value at R ∈ (0, 1) is the largest bit-error rate p that the codes in C can tolerate with a negligible probability of maximum-likelihood decoding error across a binary symmetric channel. We show that, if C is the class of cycle codes of graphs, then θC(R) ≤ ((1 - √R)2/2(1 + R)) for each R, and show that equality holds only when R is asymptotically achieved by the cycle codes of regular graphs.

This paper is about localising at night in urban environments using vision. Despite it being dark exactly half of the time, surprisingly little attention has been given to this problem. A defining aspect of night-time urban scenes is the presence and effect of artificial lighting - be that in the form of street or interior lighting through windows. By building a model of the environment which includes a representation of the spatial location of every light source, localisation becomes possible using monocular cameras. One of the challenges we face is the gross change in light appearance as a function of distance due to flare, saturation and bleeding - city lights certainly do not appear as point features. To overcome this, we model the appearance of each light as a function of vehicle location, using this to inform our data-association decisions and to regularise the cost function which is used to infer vehicle pose. In this way we develop a place-dependent but stable sensor model which is customised for the particular environment in which we are operating. We demonstrate that our system is able to localise successfully at night over 12 km in situations where a traditional point feature based system fails.

The growth-rate function for a minor-closed class $\mathcal{M}$ of matroids is the function $h$ where, for each nonnegative integer $r$, $h(r)$ is the maximum number of elements of a simple matroid in $\mathcal{M}$ with rank at most $r$. The growth-rate theorem of Geelen, Kabell, Kung, and Whittle shows, essentially, that the growth-rate function is always either linear, quadratic, exponential with some prime power $q$ as the base, or infinite. Moreover, if the growth-rate function is exponential with base $q$, then the class contains all GF$(q)$-representable matroids, and so $h(r)\ge \frac{q^r-1}{q-1}$ for each $r$. We characterise the classes that satisfy $h(r) = \frac{q^r-1}{q-1}$ for all sufficiently large $r$. As a consequence, we determine the eventual value of the growth-rate function for most classes defined by excluding lines, free spikes, or free swirls.

A matroid is GF$(q)$-regular if it is representable over all proper superfields of the field GF$(q)$. We show that, for highly connected matroids having a large projective geometry over GF$(q)$ as a minor, the property of GF$(q)$-regularity is equivalent to representability over both GF$(q^2)$ and GF$(q^t)$ for some odd integer $t \ge 3$. We do this by means of an exact structural description of all such matroids.

Let C = (C1, C2, ...) be a sequence of codes such that each Ciis a linear [ni, ki, di]-code over some fixed finite field F, where niis the length of the code words, kiis the dimension, and diis the minimum distance. We say that C is asymptotically good if, for some ε > 0 and for all i ∈ ℤ>0, we have ni≥ i and min(ki/ni, di/ni) ≥ ε. Sequences of asymptotically good codes exist. We prove that if C is a class of GF(pn)-linear codes (where p is prime and n ≥ 1), closed under puncturing and shortening, and if C contains an asymptotically good sequence, then C must contain all GF(p)-linear codes. Our proof relies on a powerful new result from matroid structure theory.