Artinian level algebras of codimension 3

In this paper, we continue the study of which $h$-vectors $\H=(1,3,..., h_{d-1}, h_d, h_{d+1})$ can be the Hilbert function of a level algebra by investigating Artinian level algebras of codimension 3 with the condition $\beta_{2,d+2}(I^{\rm lex})=\beta_{1,d+1}(I^{\rm lex})$, where $I^{\rm lex}$ is the lex-segment ideal associated with an ideal $I$. Our approach is to adopt an homological method called {\it Cancellation Principle}: the minimal free resolution of $I$ is obtained from that of $I^{\rm lex}$ by canceling some adjacent terms of the same shift. We prove that when $\beta_{1,d+2}(I^{\rm lex})=\beta_{2,d+2}(I^{\rm lex})$, $R/I$ can be an Artinian level $k$-algebra only if either $h_{d-1}<h_d<h_{d+1}$ or $h_{d-1}=h_d=h_{d+1}=d+1$ holds. We also apply our results to show that for $\H=(1,3,..., h_{d-1}, h_d, h_{d+1})$, the Hilbert function of an Artinian algebra of codimension 3 with the condition $h_{d-1}=h_d<h_{d+1}$, (a) if $h_d\leq 3d+2$, then $h$-vector \H cannot be level, and (b) if $h_d\geq 3d+3$, then there is a level algebra with Hilbert function \H for some value of $h_{d+1}$.Comment: 15 page

Generic Initial Ideals And Graded Artinian Level Algebras Not Having The Weak-Lefschetz Property

We find a sufficient condition that \H is not level based on a reduction number. In particular, we prove that a graded Artinian algebra of codimension 3 with Hilbert function $\H=(h_0,h_1,..., h_{d-1}>h_d=h_{d+1})$ cannot be level if $h_d\le 2d+3$, and that there exists a level O-sequence of codimension 3 of type \H for $h_d \ge 2d+k$ for $k\ge 4$. Furthermore, we show that \H is not level if $\beta_{1,d+2}(I^{\rm lex})=\beta_{2,d+2}(I^{\rm lex})$, and also prove that any codimension 3 Artinian graded algebra $A=R/I$ cannot be level if \beta_{1,d+2}(\Gin(I))=\beta_{2,d+2}(\Gin(I)). In this case, the Hilbert function of $A$ does not have to satisfy the condition $h_{d-1}>h_d=h_{d+1}$. Moreover, we show that every codimension $n$ graded Artinian level algebra having the Weak-Lefschetz Property has the strictly unimodal Hilbert function having a growth condition on $(h_{d-1}-h_{d}) \le (n-1)(h_d-h_{d+1})$ for every $d > \theta$ where $$h_0...>h_{s-1}>h_s.$$ In particular, we find that if $A$ is of codimension 3, then $(h_{d-1}-h_{d}) < 2(h_d-h_{d+1})$ for every $\theta< d <s$ and $h_{s-1}\le 3 h_s$, and prove that if $A$ is a codimension 3 Artinian algebra with an $h$-vector $(1,3,h_2,...,h_s)$ such that h_{d-1}-h_d=2(h_d-h_{d+1})>0 \quad \text{and} \quad \soc(A)_{d-1}=0 for some $r_1(A)<d<s$, then $(I_{\le d+1})$ is $(d+1)$-regular and \dim_k\soc(A)_d=h_d-h_{d+1}.Comment: 25 page

Hierarchically Clustered Representation Learning

The joint optimization of representation learning and clustering in the embedding space has experienced a breakthrough in recent years. In spite of the advance, clustering with representation learning has been limited to flat-level categories, which often involves cohesive clustering with a focus on instance relations. To overcome the limitations of flat clustering, we introduce hierarchically-clustered representation learning (HCRL), which simultaneously optimizes representation learning and hierarchical clustering in the embedding space. Compared with a few prior works, HCRL firstly attempts to consider a generation of deep embeddings from every component of the hierarchy, not just leaf components. In addition to obtaining hierarchically clustered embeddings, we can reconstruct data by the various abstraction levels, infer the intrinsic hierarchical structure, and learn the level-proportion features. We conducted evaluations with image and text domains, and our quantitative analyses showed competent likelihoods and the best accuracies compared with the baselines.Comment: 10 pages, 7 figures, Under review as a conference pape