arXiv subjects
Publications and source records attributed to Sean Jaffe.
For $q> p$, let $T(n,q,p)$ be the minimum number of translates of the cube \(\{0,1,\dots,p-1\}^n\) required to cover the $n$-dimensional torus $(\mathbb{Z}/q\mathbb{Z})^n$. We show that for each $q$ there exists a constant $1\le \Lambda_q \le 2$ such that $T(n,q,2)=(\Lambda_q + o(1))(q/2)^n$.
We prove that every idempotent Schur multiplier is a finite signed sum of contractive idempotent Schur multipliers. This was conjectured by Katavolos and Paulsen in 2003 and previously known only for translation-invariant Schur multipliers, by the Cohen-Host idempotent theorem. Concretely, we show that any boolean matrix $A$ with Schur multiplier norm at most~$\gamma$ (or equivalently $\lVert A\rVert_{\gamma_2} \le \gamma$) can be written as \[ A=\sum_{i=1}^{L}\sigma_i B_i,\] where $L\leq 2^{C\gamma^6}$ for an absolute constant $C$, $\sigma_i\in\{-1,1\}$ are signs, and each $B_i$ is a contractive idempotent Schur multiplier, that is, a boolean matrix whose $1$-entries form a union of all-one rectangular blocks, with no two blocks sharing a row or a column.
What is the smallest size of a family of subsets of $[n]$ such that it does not contain an induced copy of $Q_2$ as a poset (known as the \textit{diamond}), but adding a new set creates such a copy? It is easy to see that a maximal chain has this property, and thus the answer is at most $n+1$. Despite the simplicity of the diamond structure, the lower bound stagnated at $\sqrt n$ for quite some time, until recently the authors obtained a linear lower bound. In this paper, we fully solve this question showing that such a family must have size at least $n+1$.
Given a finite poset $\mathcal P$, the hypercube-height, denoted by $h^*(\mathcal P)$, is defined to be the minimum $h$ such that there exists a natural number $n$ for which the subsets of $[n]$ of size at most $h$ contain an induced copy of $\mathcal P$. The hypercube-width, denoted by $w^*(\mathcal P)$, is the smallest $w$ such that the subsets of $[w]$ of size at most $h^*(\mathcal P)$ contain an induced copy of $\mathcal P$. In other words, $h^*(\mathcal P)$ asks how `low' can a poset be embedded, and $w^*(\mathcal P)$ asks for the first hypercube in which such an `optimal' embedding occurs. These notions were introduced by Bastide, Groenland, Ivan and Johnston in connection to upper bounds for the poset saturation numbers. While it is not hard to see that $h^*(\mathcal P)\leq |\mathcal P|-1$ (and this bound can be tight), the hypercube-width has proved to be much more elusive. It was shown by the authors mentioned above that $w^*(\mathcal P)\leq|\mathcal P|^2/4$, but they conjectured that in fact $w^*(\mathcal P)\leq |\mathcal P|$ for any finite poset $\mathcal P$. In this paper we prove this conjecture. The proof uses Hall's theorem for bipartite graphs as a precision tool for modifying an existing copy of our poset.
Given a finite poset $\mathcal P$, we say that a family $\mathcal F$ of subsets of $[n]$ is $\mathcal P$-saturated if $\mathcal F$ does not contain an induced copy of $\mathcal P$, but adding any other set to $\mathcal F$ creates an induced copy of $\mathcal P$. The saturation number of $\mathcal P$ is the size of the smallest $\mathcal P$-saturated family with ground set $[n]$. The saturation numbers have been shown to exhibit a dichotomy: for any poset, the saturation number is either bounded, or at least $2\sqrt n$. The general conjecture is that in fact, the saturation number for any poset is either bounded, or at least linear. The linear sum of two posets $\mathcal P_1$ and $\mathcal P_2$, dented by $\mathcal P_1*\mathcal P_2$, is defined as the poset obtained from a copy of $\mathcal P_1$ placed completely on top of a copy of $\mathcal P_2$. In this paper we show that the saturation number of $\mathcal P_1*\mathcal A_k*\mathcal P_2$ is always at least linear, for any $\mathcal P_1$, $\mathcal P_2$ and $k\geq2$, where $\mathcal A_k$ is the antichain of size $k$. This is a generalisation of the recent result that the saturation number for the diamond is linear (in that case $\mathcal P_1$ and $\mathcal P_2$ are both the single point poset, and $k=2$). We also show that, with the exception of chains which are known to have bounded saturation number, the saturation number for all complete multipartite posets is linear.
Given a poset $\mathcal{P}$, a family $\mathcal{F}$ of elements in the Boolean lattice is said to be $\mathcal{P}$-saturated if $\mathcal{F}$ does not contain an induced copy $\mathcal P$, but every proper superset of $\mathcal{F}$ contains one. The minimum size of a $\mathcal P$-saturated family in the $n$-dimensional Boolean lattice is denoted by $sat^*(n,\mathcal{P})$.\par In this paper, we consider the poset $\mathcal V_3$ (the four element poset with one minimal element and three incomparable maximal elements) and show that $sat^*(n,\mathcal{V}_3)\geq \frac{n}{2}$. This represents the first linear lower bound for $sat^*(n,\mathcal{V}_3)$, improving upon the previously best-known bound of $2\sqrt{n}$. Our result establishes that $sat^*(n,\mathcal{V}_3) = \Theta(n)$.
For a fixed poset $\mathcal P$ we say that a family $\mathcal F\subseteq\mathcal P([n])$ is $\mathcal P$-saturated if it does not contain an induced copy of $\mathcal P$, but whenever we add a new set to $\mathcal F$, we form an induced copy of $\mathcal P$. The size of the smallest such family is denoted by $\text{sat}^*(n, \mathcal P)$.\par For the diamond poset $\mathcal D_2$ (the two-dimensional Boolean lattice), while it is easy to see that the saturation number is at most $n+1$, the best known lower bound has stayed at $O(\sqrt n)$ since the introduction of the area of poset saturation. In this paper we prove that $\text{sat}^*(n, \mathcal D_2)\geq \frac{n+1}{5}$, establishing that the saturation number for the diamond is linear. The proof uses a result about certain pairs of set systems.
We study the evolution of majority dynamics on Erd\H{o}s-R\'enyi $G(n,p)$ random graphs. In this process, each vertex of a graph is assigned one of two initial states. Subsequently, on every day, each vertex simultaneously updates its state to the most common state in its neighbourhood. If the difference in the numbers of vertices in each state on day $0$ is larger than $ \max \left\{\frac{1}{\sqrt{p}} \exp\left[A\sqrt{\log \left(\frac{1}{p}\right)}\right] , Bp^{-3/2} n^{-1/2} \right\}$ for constants $A$ and $B$, we demonstrate that the state with the initial majority wins with overwhelmingly high probability. This extends work by Linh Tran and Van Vu (2023), who previously considered this phenomenon. We also study majority dynamics with a random initial assignment of vertex states. When each vertex is assigned to a state with equal probability, we show that unanimity occurs with high probability for every $p \geq \lambda n^{-2/3}$, for some constant $\lambda$. This improves work by Fountoulakis, Kang and Makai (2020). Furthermore, we also consider a random initial assignment of vertex states where a vertex is slightly more likely to be in the first state than the second state. Previous work by Zehmakan (2018) and Tran and Vu (2023) provided conditions on how big this bias needs to be for the first colour to achieve unanimity with high probability. We strengthen these results by providing a weaker sufficient condition.
Given a finite poset $\mathcal P$, we say that a family $\mathcal F$ of subsets of $[n]$ is $\mathcal P$-saturated if $\mathcal F$ does not contain an induced copy of $\mathcal P$, but adding any other set to $\mathcal F$ creates an induced copy of $\mathcal P$. The saturation number of $\mathcal P$ is the size of the smallest $\mathcal P$-saturated family with ground set $[n]$. The saturation number for posets is known to exhibit a dichotomy: it is either bounded or it has at least $\sqrt n$ rate of growth. Determining which posets have bounded saturation number is a major open problem. In this paper we consider a `gluing' operation, formed from two finite posets $\mathcal P$ and $\mathcal Q$ by setting all elements of $\mathcal P$ to be below all elements of $\mathcal Q$. We show that (under some mild assumptions) this operation preserves bounded and unbounded saturation number. This is the first such `new from old' poset construction to be found. As an application, we show that for any poset $\mathcal P$ one may add at most 3 elements to $\mathcal P$ to obtain a poset whose saturation number growth is at most linear: this may be viewed as a step towards the other major open problem in the area, namely the conjecture that every finite poset has this growth at most linear. We also consider the poset equivalent of weak saturation for graphs: for each finite poset $\mathcal P$, we determine exactly the minimum size of a percolating family for $\mathcal P$.
Global stability and robustness guarantees in learned dynamical systems are essential to ensure well-behavedness of the systems in the face of uncertainty. We present Extended Linearized Contracting Dynamics (ELCD), the first neural network-based dynamical system with global contractivity guarantees in arbitrary metrics. The key feature of ELCD is a parametrization of the extended linearization of the nonlinear vector field. In its most basic form, ELCD is guaranteed to be (i) globally exponentially stable, (ii) equilibrium contracting, and (iii) globally contracting with respect to some metric. To allow for contraction with respect to more general metrics in the data space, we train diffeomorphisms between the data space and a latent space and enforce contractivity in the latent space, which ensures global contractivity in the data space. We demonstrate the performance of ELCD on the high dimensional LASA, multi-link pendulum, and Rosenbrock datasets.
Compressing large neural networks with minimal performance loss is crucial to enabling their deployment on edge devices. (Cho et al., 2022) proposed a weight quantization method that uses an attention-based clustering algorithm called differentiable $k$-means (DKM). Despite achieving state-of-the-art results, DKM's performance is constrained by its heavy memory dependency. We propose an implicit, differentiable $k$-means algorithm (IDKM), which eliminates the major memory restriction of DKM. Let $t$ be the number of $k$-means iterations, $m$ be the number of weight-vectors, and $b$ be the number of bits per cluster address. IDKM reduces the overall memory complexity of a single $k$-means layer from $\mathcal{O}(t \cdot m \cdot 2^b)$ to $\mathcal{O}( m \cdot 2^b)$. We also introduce a variant, IDKM with Jacobian-Free-Backpropagation (IDKM-JFB), for which the time complexity of the gradient calculation is independent of $t$ as well. We provide a proof of concept of our methods by showing that, under the same settings, IDKM achieves comparable performance to DKM with less compute time and less memory. We also use IDKM and IDKM-JFB to quantize a large neural network, Resnet18, on hardware where DKM cannot train at all.