A single complex parameter jointly tunes polarization and spatial structure of light while preserving total angular momentum symmetry.

URL https://www.sciencedirect.com/ science/article/pii/S019688582400126X

doi: https://doi.org/10.1016/j.aam.2024.102794. URL https://www.sciencedirect.com/ science/article/pii/S019688582400126X. Bao Duong and Thi Kim Hue Nguyen. Normalizing flows for conditional independence testing. Knowledge and Information Systems, 2024. Frederick Eberhardt. Almost optimal intervention sets for causal discovery. In Proceedings of the Twenty...

doi:10.1016/j.aam.2024.102794 2024

Shohei Shimizu

URL https://openreview.net/forum?id=opAU0pYlcP. Shohei Shimizu. Statistical causal discovery: LiNGAM approach. Springer, 2022. Shohei Shimizu and Kenneth Bollen. Bayesian estimation of causal direction in acyclic structural equation models with individual-specific confounder variables and Non-Gaussian distributions. Journal of Machine Learning Research-JM...

Pith/arXiv arXiv 2022

24 Published as a conference paper at ICLR 2026

m = n. 24 Published as a conference paper at ICLR 2026

2026

Every column of A is proportional to some column of B, and vice versa

⊆” direction, we just consider for each Z ∈ Ind([Q1|Q2]), its matched sources in N1 ∪ N2 can be split back into N1 and N2. For the “ ⊇

Every component of S follows a non-Gaussian distribution1. Proposition 1 (Graphical condition for irreducibility). A model (G, X) is irreducible, if and only if for each non-empty set l ⊆ L, | chG(l) \ l| ≥ 2, i.e., it has more than one child outside. Proof of Proposition 1. Due to the identifiability of OICA, irreducibility is equivalent to that there ar...

2004

The minimum-sized new cocircuits are all particular solutions. Moreover, they are exactly those minimal-inclusion ones among all solutions, which have a same (minimum) size: minimal(colaug(Q, x)) = minimum(colaug( Q, x)) = minimum(diffcc( Q, x)). (B.16)

Non-minimum-sized new cocircuits are not solutions, that is, (diffcc(Q, x) \ minimum(diffcc(Q, x))) ∩ colaug(Q, x) = ∅. (B.17)

matroid-preserving column augmentations

The intersection of minimum-sized new cocircuits may not be a solution itself, but it charac- terizes exactly items that must appear in all solutions: \ minimum(diffcc(Q, x)) = \ colaug(Q, x). (B.18) Roughly speaking, Corollary 2 takes a complement to Corollary 1: for any valid solution, it must complete new bases (witnessed by minimal new cocircuits), so...

2026

bases(QR,:) = bases(HR,:)

mrank(QR,:) = mrank(HR,:)

up to L-relabeling

mrank(QR\{Vj },Y ) < mrank(QR,Y ), that is, Vj is a coloop among R in the transversal matroid induced by QR,Y , and so removing it from ground set lowers the rank (by1). Proof of Lemma 13. We first have two immediate observations. (i) By construction,QR,{Vi} is the zero column, whereas HR,{Vi} has a single 1 in row Vj. (ii) For any Z ⊆ [m] with Vj /∈ Z, t...

2026

Here, there are 3, 3, 4 digraphs within each such class, respectively

Partitioned by cycle reversals (removing the dashed edges in the right of Figure 5), the classes connected by only edge additions/deletions (solid edges) are not necessarily isomorphic to each other. Here, there are 3, 3, 4 digraphs within each such class, respectively

same adjacencies and v-structures

To illustrate cases where cycles intersect. 36 Published as a conference paper at ICLR 2026 C.3 A P RESENTATION OF THE EQUIVALENCE CLASS In the main text, we have presented both a graphical criterion to check for equivalence (Theorem 2) and a transformational characterization to traverse the entire equivalence class (Theorem 3). These results are analogou...

2026

For every H ∈ F (G), the edge set of H is a subset of the edge set of CP(G)

The intersection of the edge sets of all H ∈ F (G) equals the solid edges of CP(G)

at least 2 and at most 4 edges from vertices Y to vertices Z

For every H ∈ E (G), let CP(H) be its own presentation. Then, CP(H) can be transformed into CP(G) via an L-relabel and a cycle reversal (alongside the solid/dashed edge types). It is worth noting that a dashed edge in a presentation means that there exists at least one equivalent digraph without this edge. However, it does not imply that dashed edges can ...

1996

Covered edge reversal

for unconditional equivalence “Covered edge reversal” (Zhang & Spirtes, 2005; Tian, 2005; Ogarrio et al., 2016; Claassen & Bucur, 2022) “Admissible edge additions/deletions and cycle reversals” (Theorem 3) Traversal algorithms DAGs traversal within one CPDAG (Meek, 1995; Chickering, 1995; Wienöbst et al., 2023); CPDAGs traversal (Steinsky, 2003; Chen et a...

2005

However, they do not directly lead to equivalence class traversal methods

Structural characterizations, which provide conditions for determining equivalence be- tween given graphs, and give rise to summary presentations. However, they do not directly lead to equivalence class traversal methods. They can be further stratified by their complexity, informativeness, or purpose, as follows: 40 Published as a conference paper at ICLR...

2026

However, as a comple- ment, they are not suited for directly determining equivalence between given graphs, or for developing summary presentations to the equivalence class

Transformational characterizations, which provide natural ways for traversing the equiva- lence class, and are useful for developing score-based algorithms. However, as a comple- ment, they are not suited for directly determining equivalence between given graphs, or for developing summary presentations to the equivalence class

trek-separation,

Traversal algorithms, for enumerating, sampling, or counting elements of the equivalence class, where transformational characterizations are usually helpful. We present in Table 2 a side-by-side overview of representative prior works and this work across these approaches in different settings. This unified view may help to better understand the contributi...

1996

For example, the two largest banks, HSBC Holdings and Hang Seng Bank, together form a 2-cycle that has 9 children across sectors, but there are no edges into them

Large banks seem to be major upstream causes. For example, the two largest banks, HSBC Holdings and Hang Seng Bank, together form a 2-cycle that has 9 children across sectors, but there are no edges into them

For example, Cheung Kong has 10 parents, but only 1 edge pointing out from it

Real estates, in contrast, seem to be downstream effect receivers. For example, Cheung Kong has 10 parents, but only 1 edge pointing out from it

For example, among 17 simple cycles in the graph, CLP Holdings belongs to 11 of them

Utilities are heavily involved in cycles. For example, among 17 simple cycles in the graph, CLP Holdings belongs to 11 of them. These cycles are often across sectors as utilities - real estate - commerce - utilities

It has one parent HSBC Holdings, and three children (all with solid edges): Cheung Kong, Hutchison, and Swire Pacific ’A’

One latent variable seems interpretable. It has one parent HSBC Holdings, and three children (all with solid edges): Cheung Kong, Hutchison, and Swire Pacific ’A’. Among them, Cheung Kong and Hutchison were two core holdings of a same group

Stocks under the same sector tend to be connected more closely. 48