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zzk231Ms1Ih	A Theory of Tournament Representations	Real-world tournaments are almost always intransitive. Recent works have noted that parametric models which assume $d$ dimensional node representations can effectively model intransitive tournaments. However, nothing is known about the structure of the class of tournaments that arise out of any fixed $d$ dimensional representations. In this work, we develop a novel theory for understanding parametric tournament representations. Our first contribution is to structurally characterize the class of tournaments that arise out of $d$ dimensional representations. We do this by showing that these tournament classes have forbidden configurations that must necessarily be a union of flip classes, a novel way to partition the set of all tournaments. We further characterize rank $2$ tournaments completely by showing that the associated forbidden flip class contains just $2$ tournaments. Specifically, we show that the rank $2$ tournaments are equivalent to locally transitive tournaments. This insight allows us to show that the minimum feedback arc set problem on this tournament class can be solved using the standard Quicksort procedure. We also exhibit specific forbidden configurations for rank $4$ tournaments. For a general rank $d$ tournament class, we show that the flip class associated with a coned-doubly regular tournament of size $\mathcal{O}(\sqrt{d})$ must be a forbidden configuration. To answer a dual question, using a celebrated result of Froster, we show a lower bound of $\Theta(\sqrt{n})$ on the minimum dimension needed to represent all tournaments on $n$ nodes. For any given tournament, we show a novel upper bound on the smallest representation dimension that depends on the least size of the number of unique nodes in any feedback arc set of the flip class associated with a tournament. We show how our results also shed light on the upper bound of sign-rank of matrices.	Published as a conference paper at ICLR 2022 A THEORY OF TOURNAMENT REPRESENTATIONS Arun Rajkumar Indian Institute of Technology RBCDSAI, IITM Abdul Bakey Mir Indian Institute of Technology Vishnu Veerathu Cohesity Inc ABSTRACT Real world tournaments are almost always intransitive. Recent works have noted that parametric models which assume d dimensional node representations can effectively model intransitive tournaments (Rajkumar & Agarwal (2016)). How- ever, nothing is known about the structure of the class of tournaments that arise out of any ﬁxed ddimensional representations. In this work, we develop a novel theory for understanding parametric tournament representations. Our ﬁrst contri- bution is to structurally characterize the class of tournaments that arise out of d dimensional representations. We do this by showing that these tournament classes have forbidden conﬁgurations which must necessarily be union of ﬂip classes, a novel way to partition the set of all tournaments. We further characterize rank 2 tournaments completely by showing that the associated forbidden ﬂip class con- tains just 2 tournaments. Speciﬁcally, we show that the rank 2 tournaments are equivalent to locally-transitive tournaments. This insight allows us to show that the minimum feedback arc set problem on this tournament class can be solved using the standard Quicksort procedure. For a general rank dtournament class, we show that the ﬂip class associated with a coned-doubly regular tournament of size O( √ d) must be a forbidden conﬁguration. To answer a dual question, using a celebrated result of Forster & Simon (2006), we show a lower bound of Ω(√n) on the minimum dimension needed to represent all tournaments on nnodes. For any given tournament, we show a novel upper bound on the smallest representation dimension that depends on the least size of the number of unique nodes in any feedback arc set of the ﬂip class associated with a tournament. We show how our results also shed light on upper bound of sign-rank of matrices. 1 I NTRODUCTION In this work, we lay the the foundations for a theory of tournament representations. A tournament is a complete directed graph and arises naturally in several applications including ranking from pairwise preferences, sports modeling, social choice, etc. We say that a tournament T on nnodes can be represented in ddimensions if there exists a skew symmetric matrix M ∈Rn×n of rank dsuch that a directed edge from ito j is present in T if and only if Mij > 0. Real world tournaments are almost always intransitive (Tversky (1969); Klimenko (2015)) and it is not known what type of tournaments can be represented in how many dimensions. This is important to understand because of the following reason: As a modeler of preference relations using tournaments, it is often more natural to have structural domain knowledge such as ‘The tournaments under consideration do not have long cycles’as opposed to algebraic domain knowledge such as ‘The rank of the skew symmetric matrix associated with the tournaments of interest is at most k’. However, algorithms that learn rankings from pairwise comparison data typically need as input the algebraic quantity - the rank of the skew symmetric matrices associated with tournaments or equivalently the dimension where they are represented (Rajkumar & Agarwal (2016)). To bridge the gap between the structural and the algebraic world, we ask and answer two fundamental questions regarding the representations of tournaments. 1) What structurally characterizes the class of tournaments that can be represented in ddimensions? 2) Given a tournament T on nnodes, what is the minimum dimension dneeded to represent it?. 1 Published as a conference paper at ICLR 2022 … FLIP CLASS 1 FLIP CLASS 2 FLIP CLASS k Locally Transitive Transitive R- cone Forbidden2d-Representable ` FLIP CLASS 1 FLIP CLASS 2 ForbiddenRepresentable Figure 1: (Left) Partitions of the set of all tournaments on nnodes using ﬂip classes. Every shaded region is a ﬂip class partition and every circle indicates a tournament. The ﬂip class that contains the transitive tournament (Flip class 1) is precisely the set of all locally transitive tournaments. This is also the set of all tournaments that can be represented in 2 dimensions (Section 5). Every ﬂip class contains a canonical representative termed the R-cone (Section 4), indicated using the larger circle inside each ﬂip class. The tournaments that cannot be represented using ddimensions appear as union of Forbidden Flip classes (Flip class kin Figure) (Section 4). (Right) Explicit ﬂip class partition of the 4 possible non-isomorphic tournaments on 4 nodes. Tournaments in ﬂip class 1 can be represented using 2 dimensions whereas tournaments in ﬂip class 2 cannot (see Section 5). . We answer the ﬁrst question by investigating the intricate structure of the rank dtournament class via the notion of forbidden conﬁgurations. Speciﬁcally, we show that the set of forbidden conﬁg- uration for the rank dtournament class must necessarily be a union of ﬂip classes, a novel way to partition the set of all tournaments into equivalence classes. We explicitly characterize the forbidden conﬁgurations for the rank 2 tournament class and exhibit a forbidden conﬁguration for the general rank dtournament class. Speciﬁcally, we show that the rank 2 tournaments are equivalent to locally transitive tournaments, a previously studied class of tournaments (Cohen et al. (2004)). Our results throw light on the connections between transitive and locally transitive tournaments and also lets us develop a classic Quicksort based algorithm to solve that the minimum feedback arc set problem on rank 2 tournaments with O(n2) time complexity. Our results for the general rank dtournament class have connections to the classic long standing Hadamard conjecture and we discuss this as well. Figure 1 gives a glimpse of some of the main results. We answer the second question by proving lower and upper bounds on the smallest dimension needed to represent a tournament on nnodes. We exhibit a lower bound of Ω(√n) using a variation of the celebrated dimension complexity result of Forster & Simon (2006) for sign matrices. To show upper bounds, we introduce a novel parameter associated to a tournament called the Flip Feedback Node set of a Tournament. This quantity depends on the least number of unique nodes in any feedback arc set of an associated tournament class for the tournament of interest and upper bounds linearly the representation dimension of any tournament. We show how our results can be used to provide upper bounds on the classic notion of sign-rank of a matrix. Previously known upper bounds for sign rank depended on the VC dimension of the associated binary function class (Alon et al. (2016)). Our upper bounds on the other hand have a graph theoretic ﬂavour. Organization of the Paper: We discuss brieﬂy in Section 2 the foundational works this paper builds upon. We introduce necessary preliminaries in Section 3. The answer to the ﬁrst question about the structural characterization of ddimensional tournament classes span Sections 4, 5 and 6. We devote Section 7 to answer the second question about the number of dimensions needed to represent a tournament. Section 8.1 explores connections of our results to upper bounds on the sign rank of a sign matrix. Finally, we conclude in Section 10. All proofs are defered to the Appendix. 2 R ELATED WORK The work in this paper builds on several pieces of work across different domains. We summarize below the most important related works under different categories: 2 Published as a conference paper at ICLR 2022 Intransitive Pairwise Preference Models: One of the main reasons to study representations of tournaments is to model pairwise preferences. Parametric pairwise preference models that can model intransitivity have gained recent interest. Rajkumar & Agarwal (2016) develop a low rank pairwise ranking model that can model intransitivity. However, their study and results were restricted to just the transitive tournaments in these classes. A generalization of the classical Bradley-Terry-Luce model (Bradley & Terry (1952), Luce (2012)) was studied in Causeur & Husson (2005). However, no structural characterization is known. Same holds for the more recent models studied in Chen & Joachims (2016), Makhijani & Ugander (2019) and Bower & Balzano (2020). Flip Classes: The notion of ﬂip classes, a novel way to partition the set of all tournaments on n nodes, was ﬁrst introduced in Fisher & Ryan (1995). The goal however was completely different and was on studying equilibrium on certain generalized rock-paper-scissors games on tournaments. Interestingly, and perhaps surprisingly, the notion of ﬂip classes turn out to be fundamental to our study of understanding forbidden conﬁgurations of ddimensional tournament classes. Dimension Complexity and Sign Rank: Dimension complexity and sign rank of sign pattern matrices were studied in Forster (2002). These results have found signiﬁcant applications in learning theory and lower bounds in computational complexity (Forster & Simon (2006)). More recently, Alon et al. (2016) study the sign rank for function classes with ﬁxed Vapnik-Chervonenkis (VC) dimension and show upper bounds. Our upper bounds however depend on certain graph theoretic properties. 3 P RELIMINARIES Tournaments: A tournament T is a complete directed graph i.e., a graph on nnodes where for every pair of nodes (i,j) either an edge is oriented from ito jor jto i. The number of nodes nin T will be usually clear from the context or will be explicitly speciﬁed. For nodes iand jin T, we say i≻T j if there is a directed edge from ito j. Given a node i, we deﬁne the out and in neighbours of ias T+ i = {j : i ≻T j}and T− i = {j : j ≻T i}respectively. Given a set of nodes S, we denote by T(S) the induced sub-tournament of T on the nodes in S. Feedback Arc Set and Pairwise Disagreement Error: Given a permutation σ on n nodes, the feedback arc set of σw.r.t the tournament T is deﬁned as Eσ(T) = {(i,j) : σ(i) >σ(j),i> T j}. The pairwise diagreement error of σ w.r.t T is given by \|Eσ(T)\| ( n 2) . It is known that ﬁnding the σ that minimizes the pairwise disagreement error w.r.t a general tournamentT is a NP-hard problem (Charbit et al. (2007)). Skew Symmetric Tournament Classes: A square matrix M ∈Rn×n is skew symmetric if Mij = −Mji∀i,j. In this paper, whenever we refer to a skew symmetric matrix M ∈Rn×n, we always assume Mij ̸= 0 ∀i̸= jand Mii = 0 ∀i. Given such an M, we denote by T{M}the tournament on nnodes induced by M where i≻T j ⇐⇒Mij >0. We refer to class of tournaments induced by rank dskew symmetric matrices as the rank dtournament class. Forbidden Conﬁgurations: A tournament class Tis a collection of tournaments. Tis said to forbid a tournament T if no tournament in T has a sub-tournament that is isomorphic to T. We call T a forbidden conﬁguration for T if T forbids T but does not forbid any sub-tournament of T. For example, the class of all acyclic/transitive tournaments has the 3-cycle as a forbidden conﬁguration i.e, the tournament T on three nodes i,j,k where i≻T j ≻T k≻T i. Representations and Tournaments: The matrix Arot ∈Rd×d is deﬁned for every even dand is a block diagonal matrix which consists ofd/2 blocks of [0−1; 1 0]. This is the canonical representation of a non-degenerate skew symmetric matrix i.e., any non-singular skew symmetric matrix can be brought to this form by a suitable basis transformation. Given a set of vectors in H = {h1,..., hn} where each hi ∈Rd for some even d, we refer to the set H as a tournament inducing representation if hT i Arothj ̸= 0 for all i ̸= j. Furthermore, we refer to the tournament induced by H as T[H] where a directed edge exits from node ito jif and only if hT i Arothj >0. It is easy to verify that hTAroth = 0 for any h ∈Rd. Any skew symmetric matrix M ∈Rn×n of rank dcan be written as M = HTArotH for some H ∈Rd×n. It follows that if M = HTArotH, then T{M}= T[H]. Positive Spans: A set of vectors H = {h1,..., hn}where each hi ∈Rd is said to positively span Rd if for any w ∈Rd, there exists non-negative constants c1,...,c n ≥0 such that ∑ icihi = w. If 3 Published as a conference paper at ICLR 2022 H positively spans Rd, then there does not exist a w ∈Rd such that wThi >0 ∀i. This is a easy consequence of Farkas’ Lemma. Remark on Notation: We reiterate that we use T(·),T{·}and T[·] to mean different objects - the tournament induced by a subset of nodes, the tournament induced by a skew symmetric matrix and the tournament induced by a representation of a set of vectors respectively. These will be usually clear from the context. 4 F LIP CLASSES , FORBIDDEN CONFIGURATIONS AND POSITIVE SPANS The main purpose of this section is understand the space of forbidden conﬁgurations of rank d tournaments. The main result of this section shows that the forbidden conﬁgurations for rank d tournament classes occur as union of certain carefully deﬁned equivalence classes of non-isomorphic tournaments. Towards this, we deﬁne the notion of ﬂip classes which was introduced ﬁrst in (Fisher & Ryan, 1995) although in a different context: Deﬁnition 1. Given a tournamentT on nnodes and a setS ⊆[n], deﬁne φS(T) to be the tournament obtained from T by reversing the orientation of all edges (i,j) such that i∈S,j ∈¯S. In other words, φS(T) is obtained from T by reversing the edges across the cut (S, ¯S) = {(i,j) : i∈S,j ∈¯S}. Deﬁnition 2. A class of tournaments on nnodes is called cut-equivalent if for every pair of tourna- ments T,T′in the class, there exists a S ⊆[n] such that T′is isomorphic to φS(T) It is easy to show that the set of cut-equivalent tournaments form an equivalence relation over the set of all tournaments on nnodes Fisher & Ryan (1995). The corresponding equivalence classes are called a ﬂip classes. We denote by F(T) the ﬂip class of T i.e., the equivalence class of all cut-equivalent tournaments to T. In the following theorem we show the fundamental relation between ﬂip classes and forbidden conﬁgurations. Theorem 1. Let T, a tournament on knodes, be a forbidden conﬁguration for some rank dtourna- ment class. Then every tournament in the ﬂip class of T is also a forbidden conﬁguration for the rank dtournament class. Corollary 1. (Structure of Forbidden conﬁgurations) Forbidden conﬁgurations of rankdtourna- ment classes are unions of ﬂip classes. Thus, to characterize the forbidden conﬁguration of rank dtournament classes, we need to understand the ﬂip classes. We begin with the following simple but useful deﬁnition. Deﬁnition 3. A tournament T is called a R-cone if there exists a node i∗such that • i∗≻T jfor all j ̸= i∗( j ≻T i∗for all j ̸= i∗) • T(T+ i∗) ( T(T− i∗) ) is isomorphic to the tournament R. R-cones are essentially the tournament R along with an additional node that either beats or loses to nodes in R. R-cones are useful as they be viewed in some sense as canonical tournaments in ﬂip classes. This is justiﬁed because of the following observation. Proposition 1. Every ﬂip class contains an R-cone for some tournament R. The above observation says that to identify the forbidden conﬁgurations for a given tournament class, it sufﬁces to identify all forbidden R-cones. Then by Corollary 1, the associated ﬂip classes will be the set of all forbidden conﬁgurations. However, it does not throw light on what property the tournament R must satisfy. The following lemma establishes this. Lemma 2. Let R be a tournament with the property that if R = R[H] for some representation H = {h1,..., hn} ∈Rd then H positively spans Rd. Then rank d tournament class forbids R-cones. The above lemma is extremely helpful in the sense that it reduces the study of forbidden conﬁgurations to the study of ﬁnding tournaments such that any representation that induces it must necessarily 4 Published as a conference paper at ICLR 2022 positively span the entire Euclidean space. Note that there may be several representations which positively span the entire space. This does not mean their the associated coned tournaments are forbidden conﬁgurations. Instead, we start with an R cone and conclude it is forbidden if every representation that induces R necessarily positively spans the entire space. It is a non-trivial problem to identify such tournaments R for an arbitrary dimension d. In the following sections, we explicitly identify the only forbidden ﬂip class for rank 2 tournaments, one forbidden ﬂip class for rank 4 and then (a potentially weaker) forbidden ﬂip class for the general rank dcase. 5 R ANK 2 TOURNAMENTS ⇐⇒LOCALLY TRANSITIVE TOURNAMENTS The goal of this section is to characterize the forbidden conﬁgurations of rank 2 tournaments. Thanks to Lemma 2, this reduces to the problem of identifying a tournament whose representation necessarily span the entire space. The following lemma exhibits this tournament. Lemma 3. Let H = {h1,h2,h3}∈ R2 be two dimensional representation of 3 nodes which induce a 3 cycle tournament. Then the set H positively spans R2. Furthermore, the 3 cycle is the only such tournament on 3 nodes. The above lemma immediately implies that a coned 3-cycle is the only forbidden conﬁguration for rank 2 matrices. This is indeed true. However, for the purposes of generalizing our result (which as we will see will be useful when discussing the higher dimension case), we will view the 3-cycle as a special case of a doubly regular tournament (deﬁned next). Deﬁnition 4. A tournament T on nnodes is said to be n-doubly regular if \|T+ i \|= \|T+ j \|for all i,j and \|T+ i ∩T+ j \|= kfor some ﬁxed kfor all i̸= j Trivially the 3-cycle is the only 3-doubly regular tournament. The following lemma establishes that the ﬂip class of a coned 3 doubly regular tournament only contains itself. Proposition 2. The ﬂip class of 3-doubly-regular-cone does not contain any other tournament. Theorem 4. 3-doubly-regular-cone is the only forbidden conﬁguration for the rank 2 tournament class. We have thus far established that any rank2 tournament on nnodes forbids the 3-doubly-regular-cone. The advantage of this result is that we can go one step further and explicitly characterize the rank 2 tournament class. To do this, we need the deﬁnition of a previously studied tournament class. Deﬁnition 5. A tournament T is called locally transitive if for every node iin T, both T(T+ i ) and T(T− i ) are transitive tournaments Before seeing why locally transitive tournaments are relevant to our study, we ﬁrst show that they are intimately connected to transitive tournaments via the following characterization. Theorem 5. (Connection between Transitive and Locally Transitive Tournaments) The set of all non isomorphic locally transitive tournaments on n nodes is equivalent to the ﬂip class of the transitive tournament on nnodes. We will see next that this key result allows us to immediately characterize the rank 2 tournament class. We will see later (Section 7) that this result is also crucial in determining an upper bound on the dimension needed to represent a given tournament. Theorem 6. (Characterization of rank 2 tournaments) A tournament T on n nodes is locally transitive if and only if there exists a skew symmetric matric M ∈Rn×n with rank(M) = 2 such that the T{M}= T. It is perhaps surprising that a purely structural description of a tournament class namely that of local transitivity turns out to be exactly equivalent to the rank 2 tournament class. To the best of our knowledge, this characterization appears novel and hasn’t been previously noticed. One of the interesting consequence of the above characterization is that the minimum feedback arc set problem on rank 2 tournaments can be solved using a standard quick sort procedure. This is formalized below. 5 Published as a conference paper at ICLR 2022 Theorem 7. (Minimum Feedback Arc Set is Poly-time Solvable for Rank 2 tournaments) Let T be a locally transitive tournament on nnodes and let σ1 be the permutation returned by running a standard quick sort algorithm choosing 1 as the initial pivot node and where the outcome of comparison between items iand j is iif and only if i≻T j. Let σk be obtained from σ1 by k−1 clockwise cyclic shifts for k ∈[n]. Let Ek be the feedback arc set of σk w.r.tT. Then mink\|Ek\| achieves the minimum size of the feedback arc set for T. Proof. (Sketch) The proof involves two steps: ﬁrst arguing that by ﬁxing any pivot, quick sort would return a ranking that is a cyclic shift of σ1. The second step involves inductively arguing that one of the cyclic shift must necessarily minimize the feedback arc set. 5.1 R ANK 4 TOURNAMENTS We now turn to rank 4 tournaments. We could have directly considered rank dtournaments, but it turns out that what we can show a slightly stronger result for rank 4 tournaments than the general case and so we focus on them separately. While it is arguably simple in the rank 2 case to identify the tournament that necessitates the positive spanning property, it is not immediately clear in the rank 4 case. A ﬁrst guess would be to consider the regular tournaments (as the 3 cycle for rank 2 is also a regular tournament) on 5 nodes or 7 nodes. However, these turn out to be insufﬁcient as one can construct counter examples of regular tournaments on up to 7 nodes with representations that don’t span the entire R4. In fact, as we had deﬁned earlier, the right way to generalize to higher dimension turns out to be using doubly regular tournaments. Theorem 8. The 11-doubly-regular-cone is a forbidden conﬁguration for rank 4 tournament class. Note that while for the rank 2 case, we were able to prove that the only forbidden ﬂip class is the one that contains a coned 3 cycle, we have not shown that the only forbidden conﬁguration for rank 4 class is the 11 doubly regular cone. In fact, we believe that the smallest forbidden tournament for rank 4 class is the 7- doubly regular cone. However, we haven’t been able to prove this. This appears to be a non-trivial problem. From our simulation experiments, we observe that the only ﬂip class that could be forbidden on 8 nodes is the one that contains the 7-doubly regular cone. In particular, we were able to produce examples of representations for all other ﬂip classes on 8 nodes. However, this does not imply that the ones where we could not produce a forbidden conﬁguration is in fact forbidden. Unfortunately, it seems tricky to prove this and we don’t have a way to show this at this point. On the other hand, as we will see in the next section, the result in Theorem 8 is still stronger than the result for the general rank dtournaments. Having discussed rank 2 and rank 4 cases separately, we next turn our attention to the general rank d tournament class. 6 R ANK d TOURNAMENTS AND THE HADAMARD CONJECTURE From the understanding of rank 2 and rank 4 tournament classes in the previous sections, and noting that the corresponding forbidden conﬁgurations are intimately related to doubly regular tournaments, it is tempting to conjecture that this is true in general. Conjecture 1. Rank 2dtournament class forbids 4d−1-doubly-regular-cones. Ideally, the conjecture above must have a qualiﬁer ‘if they exist’ for the 4d−1 doubly regular cone. This is because of the equivalence between doubly regular tournaments on 4d−1 nodes and Hadamard matrices in {+1,−1}4d×4d (Reid & Brown (1972)). A matrix H ∈{+1,−1}n×n is called Hadamard if H′H = nI where I is the identity matrix. It is known that there is a bijection between skew Hadamard matrices and doubly regular tournaments Reid & Brown (1972). A long standing unsolved conjecture about Hadamard matrices is the following: Conjecture 2. (Hadamard) There exists a Hadamard matrix of order4dfor every d> 0. If Conjecture 1 were true, then it would imply the existence of 4d−1-doubly regular tournament for every dand thus would imply the Hadamard conjecture is true. In fact, Conjecture 1 being true would say more which we state below: Conjecture 3. There exists a skew symmetric Hadamard matrix of order4dfor every d> 0. 6 Published as a conference paper at ICLR 2022 The main result of this section is a weaker form of the conjecture: Theorem 9. Rank 2(d−1) tournament class forbids 12d2 −1-doubly-regular-cones if they exist. 7 H OW MANY DIMENSIONS ARE NEEDED TO REPRESENT A TOURNAMENT ? The previous sections considered a speciﬁc rank dtournament class and tried to characterize them using forbidden conﬁgurations. In this section, We turn to the dual question of understanding the minimum number of dimensions needed to embed a tournament. We start by not considering a single tournament T but the set of all tournaments on nnodes. We show below a general result which provides a lower bound on the minimum dimension needed to embed any tournament on nnodes. Theorem 10. (Lower Bound on minimum representation dimension) Let T be a tournament on n nodes. Then there existsH = {h1,...,h n}∈ Rdsuch that T = T[H] only if d∑n i=1(ρi(T+I))2 ≥ n2. Furthermore, let dbe the minimum dimension needed to embed every tournament on nnodes. Then, d≥√n. The result follows the arguments in the celebrated work of Forster & Simon (2006). As Alon et al. (2016) point out, Forster’s technique cannot be stretched further in obvious ways to get upper bounds. The above theorem tells us that in the worst case at leastΩ(√n) dimensions are necessary to represent all tournaments on nnodes. In fact if Conjecture 1 were true, the minimum dimension would be Ω(n). However, in practice one might not encounter tournaments with such extremal/worst-case properties. Remark: The lower bound for the representation dimension of random tournaments (where the orientation of each edge is determined by an unbiased Bernoulli coin toss) will depend on the singular values of random tournaments. However, we don’t expect the representation dimension to be of independent of n, the number of nodes. Loosely speaking, a doubly regular tournament is "like" a random tournament (the associated Hadamard Tournament has been used in deterministic perturbation schemes as alternatives for random sign matrices (Bhatnagar et al. (2003))). On the other hand, the most interesting real-world tournaments might be characterized by constant sized node representations and hence may be structurally much more constrained than random tournaments. Typically, a smaller number of dimensions might be enough to represent tournaments of practical interest. Our goal below is to upper bound on the number of dimensions needed to embed a given tournament T. Recall that Eσ(T) denotes the feedback arc set of a permutation σ w.r.t a tournament T. We deﬁne the number of nodes involved in the feedback arc set as follows: θ(σ,T) = \|{i: ∃j : σ(i) >σ(j),(i,j) ∈Eσ(T)}\|. We next deﬁne a crucial quantity µ(T) which we term as the Flip Feedback Node set size. This quantity will determine an upper bound on the dimension where a tournament can be represented: Deﬁnition 6. Given a tournament T, deﬁne the Flip Feedback Node Set Size as follows: µ(T) = min σ min T′∈F(T) θ(σ,T′) In words, given a tournament T, the quantity µ(T) captures the minimum number of nodes involved in any feedback arc set among all tournaments in the ﬂip class of T. For instance if T is a locally transitive tournament then µ(T) would be 0 - as T is necessarily in the ﬂip class of a transitive tournament and so the Eσ corresponding to the topological ordering of the transitive tournament would have an empty feedback arc set. As another example, consider T to be the coned 3-cycle. The ﬂip class of this tournament contains only itself and the best permutation will have one edge in the feedback arc set. Thus µ(T) = 1. In general, it is trivially true that µ(T) is upper bounded by n, the number of nodes. Howeverµ(T) could be much smaller than ndepending on T. The main result of this section is the theorem below that shows that µ(T) gives an upper bound on the number of dimension needed to represent any tournament. Theorem 11. (Upper Bound on minimum representation dimension) For any tournamentT, there exists H = {h1,..., hn}∈ R2(µ(T)+1) such that T = T[H]. 7 Published as a conference paper at ICLR 2022 1 2 3 4 5 6 ` Figure 2: A simple tournament to illustrate that the quantity µ(T) need not be same as the size of the minimum feedback arc set. All edges go from left to right except the ones in red. See Remark 2 in Section 7 for details. Remark 1: The above theorem says that one can always obtain an representation H in dimension d= O(µ(T)) that induces T. The bound gets tighter for tournaments with smaller feedback arc sets, which is what one might typically expect in practice. Note that even for some tournaments that may have a large feedback arc set, the associated ﬂip class might contain a tournament with a smaller feedback arc set. Remark 2: We note that in general µ(T) is not necessarily the cardinality of the minimum feedback arc set among all tournaments in the ﬂip class of T. Instead µ(T) captures the cardinality of the set of nodes involved in any feedback arc set. To see why these two could be different, consider the tournament in Figure 2. Here, σa = [6 1 2 3 4 5] is the permutation minimizing the feedback arc set. Let σb = [1 2 3 4 5 6]. Note that \|Eσa\|= 2, θ(σa,T) = \|{1,2}\|= 2. However \|Eσb\|= 3, yet θ(σb,T) = \|{6}\|= 1. Thus, σb gives a tighter upper bound on the dimension needed to represent T. 8 A PPLICATIONS OF THE RESULTS 8.1 C ONNECTIONS TO SIGN RANK The sign rank of a matrix G ∈{+1,−1}m×n is deﬁned as the smallest integer dsuch that there exists a matrix M ∈Rm×n of rank dthat satisﬁes sign(M) = G1. Here sign(z) = 1 if z >0 and −1 otherwise. A breakthrough result on the lower bound on the sign rank was given by Forster & Simon (2006). However, good upper bounds have been harder to obtain. We show below how Theorem 11 also translates to an upper bound on the sign rank of any sign matrix G. Theorem 12. Let G ∈{+1,−1}m×n be an arbitrary sign matrix. Let T be any tournament on m+ nnodes. Let S = {1,...,m }and let edge orientations of the cut(S, ¯S) in T be determined by G Then sign-rank(G) ≤2(µ(T) + 1) We argue that the the above result can give signiﬁcantly tigher bounds than the bounds in Alon et al. (2016). The simplest example one can consider is a locally transitive tournament onnnodes. Viewing this as a sign pattern matrix, it is easy to argue that the VC dimension of such a matrix is atmost 2. Theorem 5 in Alon et al. (2016) shows that the sign rank is upper bounded by O(√n)) when VC dimension is 2 (For a general hypothesis class of VC dimension d, the upper bound is O(n1−1 d). On the other hand, by deﬁnition µ(T) for any locally transitive tournament is exactly 0 (and so the upper bound of 2µ(T) + 1 is exactly 2 and is tight) for any locally transitive tournament which is independent of the number of nodes n. The important reason for this signiﬁcantly improved bound using µ(T) is that we consider only skew symmetric sign pattern matrices while Alon et al. (2016) look at the worst case (w.r.t representation dimension/sign rank) sign pattern matrices for a ﬁxed VC dimension. Our bound reduces the study of sign rank to a more graph theoretic study of the feedback arc set problem. It is not clear if the upper bound of 2(µ(T) + 1)can be improved and we leave this to future work. 8.2 C ONNECTIONS TO LEARNING FROM PAIRWISE COMPARISONS Consider a learning to rank problem from pairwise comparisons. Here, a set of nitems need to be ranked from a subset of pairwise comparisons among them. Every pair that is chosen for comparison is compared a ﬁxed number of times. Every time items iand jare compared, iis preferred over j 1Sign-rank can be deﬁned for any general matrix G ∈ Rm×n. We restrict to the sign matrices to make the connections to the tournament matrices explicit 8 Published as a conference paper at ICLR 2022 with probability Pij. A common and popular model to capture these probabilities is the Bradley- Terry-Luce (BTL) model Bradley & Terry (1952) wherePij = si/(si + sj) for some score vector s ∈Rn. Note that the model is completely speciﬁed by the score vector s which in turn completely determines the probability preference matrix P. The learning problem is to learn these unknown score vector s ∈Rn from noisy pairwise comparisons. Once the score vector is learnt, a ranking can be obtained by sorting these scores and the pairwise probabilities of unseen pairs of items can be predicted. The above BTL model is an example of a rank-2 model in the sense that the probability preference matrix P results in a rank 2 matrix under the log-odds skew symmetric transformation. Indeed, if we deﬁne Mij = log( Pij Pji ). then equivalently Mij = log(si) −log(sj). It is easy to show that M is a rank 2 skew symmetric matrix. However, the major disadvantage of the BTL model is that it can capture only transitive preferences i.e, Pij > 0.5 and Pjk > 0.5 = ⇒ Pki > 0.5. In real world situations, intransitivity is very common. To achieve intransitivity, the simplest way would be to start with a general rank rskew symmetric matrix M and consider the probability matrix that determines the preference probabilities as Pij = 1/(1 + exp(−Mij)). Here, the learning problem would be to estimate the two score vectors or equivalently the two dimensional representation for each item. Previous studies (Rajkumar & Agarwal (2016)) show that this can be learnt using Matrix completion based approaches or maximum likelihood based approaches (Chen & Joachims (2016)). However, what was not known earlier is the structure of tournaments that can be captured using such low rank restrictions which is important to decide on the parameter r. Our work helps modellers make informed decision to choose rbased on explicitly identifying the types of tournaments that can be modelled and subsequently learnt. We discuss next a simple application where modeling a ranking from pairwise comparisons problem can beneﬁt from the insights gained using our results. Consider a situation of modelling sports tournaments such as Tennis. Here, one can choose to model the players (nodes) using 2-dimensions where the dimensions corresponds to their offense (forehand) and defense (backhand) strengths respectively. When two players compete, the advantage of the offense strength of player1 w.r.t the defense of player 2 and vice versa determine the outcome of the match. This is precisely captured by a rank 2 model where the learning problem would be to infer these latent offense/defense strength of each player from outcomes of pairwise competitions. Theorem 6 says that such a model would immediately lead to locally transitive tournaments among the players. This structural characterisation now gives insights to the modeller if 2 dimensions are enough to model the players or not. 9 R EAL WORLD TOURNAMENT EXPERIMENTS We conducted simple experiments on real world data sets. Speciﬁcally, we considered 114 real world tournaments that arise in several applications including election candidate preferences, Sushi preferences, cars preferences, etc (source: www.preflib.org). The number of nodes in these tournaments varied from 5 to 23. Out of the 114 tournaments considered, 76(66.67%) were in fact locally transitive. For these tournaments, the upper bounds and lower bounds given by our theorems matched and was equal to 2. Interestingly, even for the non-locally transitive tournaments, the lower bound still turned out to be 2. We computed the upper bound for tournaments of size at most 9 (we did not do it for larger tournaments as this involves a brute force search) and found the value to be either 4 or 6. This shows that the upper bounds are usually non-trivial and efﬁciently approximating it is an interesting direction for future work. 10 C ONCLUSION In this work, we develop a theory of tournament representations. We show how ﬁxing the representa- tion dimension enforces, via forbidden conﬁgurations, restrictions on the type of tournaments that can be represented. We study and characterize rank 2 tournaments and show forbidden sub-tournaments for the rank d tournament class. We develop upper and lower bounds for minimum dimension needed to represent a tournament. Future work includes attempting to look deeper into some of the conjectures presented and possible strengthening of some of the bounds presented. 9 Published as a conference paper at ICLR 2022 REFERENCES Noga Alon, Shay Moran, and Amir Yehudayoff. Sign rank versus vc dimension. In Conference on Learning Theory, pp. 47–80. PMLR, 2016. Shalabh Bhatnagar, Michael C Fu, Steven I Marcus, and I-Jeng Wang. Two-timescale simulta- neous perturbation stochastic approximation using deterministic perturbation sequences. ACM Transactions on Modeling and Computer Simulation (TOMACS), 13(2):180–209, 2003. Amanda Bower and Laura Balzano. Preference modeling with context-dependent salient features. In International Conference on Machine Learning, pp. 1067–1077. PMLR, 2020. Ralph Allan Bradley and Milton E Terry. Rank analysis of incomplete block designs: I. the method of paired comparisons. Biometrika, 39(3/4):324–345, 1952. David Causeur and François Husson. A 2-dimensional extension of the Bradley–Terry model for paired comparisons. Journal of statistical planning and inference, 135(2):245–259, 2005. Pierre Charbit, Stéphan Thomassé, and Anders Yeo. The minimum feedback arc set problem is NP-hard for tournaments. Combinatorics, Probability and Computing, 16:01–04, 2007. Shuo Chen and Thorsten Joachims. Modeling intransitivity in matchup and comparison data. In Proceedings of the ninth acm international conference on web search and data mining, pp. 227–236, 2016. Nir Cohen, Marlio Paredes, and Sofía Pinzón. Locally transitive tournaments and the classiﬁcation of {(1,2)}-symplectic metrics on maximal ﬂag manifolds. Illinois Journal of Mathematics, 48(4): 1405–1415, 2004. David C Fisher and Jennifer Ryan. Tournament games and positive tournaments. Journal of Graph Theory, 19(2):217–236, 1995. Jürgen Forster. A linear lower bound on the unbounded error probabilistic communication complexity. Journal of Computer and System Sciences, 65(4):612–625, 2002. Jürgen Forster and Hans Ulrich Simon. On the smallest possible dimension and the largest possible margin of linear arrangements representing given concept classes. Theoretical Computer Science, 350(1):40–48, 2006. Alexander Y Klimenko. Intransitivity in theory and in the real world. Entropy, 17(6):4364–4412, 2015. R Duncan Luce. Individual choice behavior: A theoretical analysis. Courier Corporation, 2012. Rahul Makhijani and Johan Ugander. Parametric models for intransitivity in pairwise rankings. In The World Wide Web Conference, pp. 3056–3062, 2019. Arun Rajkumar and Shivani Agarwal. When can we rank well from comparisons of O(nlog(n)) non-actively chosen pairs? In Conference on Learning Theory, pp. 1376–1401. PMLR, 2016. KB Reid and Ezra Brown. Doubly regular tournaments are equivalent to skew hadamard matrices. Journal of Combinatorial Theory, Series A, 12(3):332–338, 1972. Amos Tversky. Intransitivity of preferences. Psychological review, 76(1):31, 1969. 10 Published as a conference paper at ICLR 2022 A A PPENDIX Proof of Theorem 1 Proof. Assume there is a T′ ∈F(T) which is not forbidden for rank dtournament class. Thus, ∃H′= {h′ 1,..., h′ n}∈ Rd such that T′= T[H′]. By deﬁnition, there must also exist a S ⊆[n] such that T′= φS(T). Consider the representation H obtained from H′where hi = −h′ i for all i ∈S and hi = h′ i otherwise. It is easy to verify that T = T[H] which is a contradiction to the assumption that T is a forbidden conﬁguration for rank dtournaments. Proof of Proposition 1 Proof. Consider any tournament T. Let T′= φ{i∪T− i }(T) for an arbitrary node i. By deﬁnition T′∈F(T). Also, T′is a R-cone, coned by i. Proof of Lemma 2 Proof. Consider any representation H that induces a tournament R. By assumption of the theorem, H positively spans Rd. Thus, by Farkas’ lemma, there cannot exist av ∈Rd such that vThi >0 ∀i. Note that if R-cone is not a forbidden conﬁguration for rank dtournaments, then there must exist a h ∈Rd such that hTArothi > 0 ∀i. As Arot is invertible, one can set v = ( Arot)Th with the property that vThi > 0 ∀i. But this contradicts the conclusion drawn earlier from Farka’s lemma. Proof of Proposition 2 Proof. This is easily veriﬁed by checking all tournaments inF(T) where T is the coned 3-cycle. Proof of Lemma 3 Proof. Wlog, assume that 1 ≥T 2 ≥T 3 ≥T 1. Then, it must be the case that the counterclockwise angle between the representation of the corresponding items must be ≤180 degrees. However, if the representations {h1,h2,h3}did not positively span R2, then by Farka’s Lemma, there must be some supporting hyperplane for the representations. However, this would imply at least one of the node pairs {(1,2),(2,3),(3,1)}must necessarily make an angle ≥180 degrees. But this contradicts the assumption that the nodes form a 3 cycle. Proof of Theorem 5 Proof. Let T be a transitive tournament on nnodes and let T′∈F(T). Then, there exists some S ⊆[n] such that T′ = φS(T). Consider any node i ∈[n]. Deﬁne following 4 subset of nodes associated with i: S1 = T+ i ∩S,S2 = T+ i \S1,S3 = T− i ∩S,S4 = T− i \S. Note that each of T(Sk) for k = 1 to 4 is a transitive sub-tournament and the relationship across Si,Sj for any two sets is either one completely beats the other or completely loses to the other. Note that in φS(T) the orientation of the edges across these sets is either ﬂipped as a whole or not ﬂipped at all. Thus, exactly two of these sets will be part of T′+ i and two part of T− i (the exact sets among these sets will depend on whether i∈Sor not), thus preserving the local transitivity property. Thus T′is locally transitive. To prove the opposite direction, letT be a locally transitive tournament. Let i∈[n] be an arbitrary node. We argue that T′:= φT+ i (T) is a transitive tournament. We will show this by arguing that there does not exist a 3 cycle in T′. Consider any 3-cycle a >T b >T c >T a. As T is locally transitive, not all {a,b,c }can be in T+ i . Also not all {a,b,c }can avoid T+ i as [n] \T+ i is transitive. Thus at least one and at most two of{a,b,c }belongs to T+ i (T). This means that the 3-cycle becomes a transitive tournament in T′:= φT+ i (T). Thus every cycle in T becomes transitive in T′. Now consider any 3 nodes which forms a transitive tournament a >T b >T cand involves at least one 11 Published as a conference paper at ICLR 2022 node and at most two nodes in T+ i . Then there are only two cases to consider: (1) a ∈[n] \T+ i and {b,c}∈ T+ i or (2) {a,b}∈ [n] \T+ i and c∈T+ i . In both these cases, it is easy to verify that the the corresponding tournament in T′is either the transitive tournament b>T′>c> T′ aor the transitive tournament c>T′ a>T′ b. As these are the only possibilities, the result follows by noting that T′= φT+ i (T) =⇒ T = φT+ i (T′) and so T ∈F(T′). Proof of Theorem 6 Proof. Assume T is locally transitive. Then by Theorem 5, it must be in the ﬂip class of some transitive tournament T′i.e. T = φS(T′) for some S ⊆[n]. It is easy to represent a transitive tournament using a rank 2 skew symmetric matrix. Indeed pick any vector u ∈ Rn which is sorted according to the topological ordering of T′. Let v ∈Rn be the all ones vector. Then M = uvT −vuT represents T′. Let M = (H′)TArotH for some H′∈R2×n.Then the columns of H′represent T′. Now consider the representation H′obtained from H′where the columns indexed by Sare multiplied by −1. This does not change the rank of H′and it can be veriﬁed that T = T[H]. To prove the other direction, consider any rank 2 skew symmetric matrix M ∈Rn×n. Then there must exist u,v ∈Rn such that M = uvT −vuT. Consider any node i∈[n]. Consider three nodes a,b,c such that uivj >ujvi for j ∈{a,b,c }. Furthermore let uavb >vaub and ubvc >vbuc. Then by carefully going over all {±}sign possibilities for {ui,ua,ub,uc,vi,va,vb,vc}, one can conclude that it must be the case that uavc >vauc. This just shows that T+ i is transitive. Analogously one can show that T− i is also transitive. As iwas arbitrary, the result follows. Proof of Theorem 7 Proof. Recall that the classic quick sort algorithm picks a pivot node (say 1) and places all nodes that beat the pivot to the right in the ranking and those thatlose to the left and then recurses on the left and right subsets. As T is locally transitive, choosing any pivot iwould correspond to ﬁxing the pivots position and simply returning the ranking [σ(N− i ),i,σ (N+ i )] where σ(N+ i ), σ(N− i ) correspond to the topological ordering of the transitive tournamentsT(N+ i ) and T(N− i ) respectively. We ﬁrst argue that changing the pivot only cyclically shifts the ﬁnal ranking. To see why this is true, consider two pivots iand jand their corresponding rankings σi and σj. Without loss of generality, assume that the ranking σi = [1,...,n ] and i>T j. We argue that there exists an integer ksuch that N+ j = {j+ 1 mod n,(j+ 2) mod n,..., (j+ k) mod n}(where by convention n mod n = n). As N+ i is transitive, and jis part of it, it must be the case that all the nodes {j+ 1,...,j + n}∈ N+ j . Then to prove the claim, it remains to be shown that the set N− i ∩N+ j is either empty or must be the nodes {1,...,ℓ }for some ℓ<i . If empty, we are done. If not, assume for the sake of contradiction that there exists three succesive integers ℓa,ℓb,ℓc < isuch that ℓa,ℓc <T j but ℓb >T j. It is easy to verify that this cannot happen as it would lead to T({ℓa,ℓb,ℓc,j}) being a forbidden conﬁguration. Notice that as every locally transitive tournament is in the ﬂip class of a transitive tournament i.e., T = φS(T) for some S, one can divide the set of all nodes into 2k+ 1 groups as follows: Let [1,...,n ] be the ordering corresponding to the transitive tournament wlog. Starting from 1, add as many nodes to a group such that all the elements belong to either S or ¯S. Once the condition is violated, create a new group and continue the same process. It is not hard to verify that 2k+ 1 groups will be formed in this process for some k ≥0. Moreover, each group would separate two other groups by construction. We can ﬁrst show that the items of a single group must appear in consecutive positions in one of the optimal rankings. This is proven as follows. Consider there exists an optimal ranking with items which belong to the same group not occurring consecutively. Consider two items belonging to the same group, which have items from other groups present in between them in the ranking. Consider these items to be a1,a2, with a1 present above in the rankings. Consider the number of upsets that the two items are involved in to be u1 and u2. If u1 ≤u2, a2 can be placed right after a1 in the ranking, creating a better or equivalent ranking in terms of upsets. Similarly if u1 ≥u2, a1 can be placed directly above a2 in the rankings to create 12 Published as a conference paper at ICLR 2022 an equivalent or better ranking. Therefore there exists an optimal ranking which has all items in the same group consecutively. This theorem is then reduced to ﬁnding a ranking of groups, which is proven using induction on k. Base Case Consider the base case with k= 1. Let there be 3 groups, C, A1, B1. We can say that the optimal ranking cannot be any of the following CB1A1 A1CB1 B1A1C since all three rankings can be made better by swapping the second and third ranked groups. Therefore the 3 possible optimal rankings are CA1B1 A1B1C B1CA1 which are cyclic shifts of each other. Inductive Step One property of rankings which is useful for the inductive step proof is as follows. Let there be2k+ 1 groups G= {g1,g2 ...g 2k+1}. Label the optimal ranking with the condition that gi be placed ﬁrst in the ranking as Ri. The ranking Ri with gi removed must be the optimal ranking for G\{gi}. This can be shown using contradiction i.e, if there was a better ranking for G\gi, that ranking with gi appended to the front would be better than Ri. We now assume the theorem is true for size 2k−1 instances and aim to prove for the same for size 2k+ 1 instances. Consider g1 as the ﬁrst group in the ranking. This creates a certain number of upsets, for the purposes of ranking the remaining groups, 2 of the remaining groups can be merged into a single group. This follows from the observation earlier that each group also ’separates’ two groups. This can be considered an instance of the size 2k−1 problem. Therefore the set of optimal rankings with g1 as the ﬁrst group in the rankings is made up of g1 as the ﬁrst group and a cyclic sweep of the remaining items to ﬁll the remaining positions. Therefore the optimal permutation must be among the sets created by considering each of the 2k+ 1 groups as the ﬁrst group in the rankings. Let Ri,j represent the ranking which has group gi as the ﬁrst group and the remaining groups present as a cyclic sweep from gj. Consider the case of R1,k. Let xi represent the number of items in group gi. If R1,k is a better ranking than Rk,k+1, it implies that n+1∑ i=k xi > 2n+1∑ i=n+2 xi (1) by considering the shift of g1 in the two rankings. The difference in the number of upsets between R1,2 and R1,k is given by x2(−xk−xk+1 ... −xn+2 + xn+3 ...x 2n+1) +x3(−xk−xk+1 ... −xn+3 + xn+4 ...x 2n+1) ... Using Equation 1, it can be seen that each of the above terms are negative for any j ≤n+ 1, making R1,2 the better ranking. Any j >n+ 1 cannot be considered as an optimal ranking since the ﬁrst group as per the ranking must precede the second(otherwise switching them would decrease the upsets). Since either R1,2 or Rk,k+1(both counterclockwise orderings) is better than R1,k whenever k≤n+ 1(R1,k cannot be the optimal ranking when k >n+ 1), and since this can be generalised for any Ri,j, it is shown that one of the counterclockwise orderings of the items is the optimal ranking. We note that the above proof also appeared in ?. However, the insights via ﬂip classes were not present there. 13 Published as a conference paper at ICLR 2022 A.0.1 F INDING FORBIDDEN CONFIGURATIONS Given a tournament and a representation dimension, there is no known method to check whether the tournament can be represented by vectors in the given dimension. The forbidden conﬁgurations presented above were found by carefully creating an exhaustive set of cases, and showing each one causes a contradiction. The techniques used are presented below. Proof of Theorem 8 Proof. Consider the representation of tournaments as given in Subsection 3. Since adding minor noise to each hi will not change tournament, we can deal only with cases in which any size 4 subset of h1,h2,...h12 consists of linearly independent vectors. By using this property, and without loss of generality, c1h1 + c2h2 + c3h3 + c4h4 = h5 (2) We can construct cases based on the signs of the coefﬁcients(ci) in the above equation. There are 24 = 16 sign patterns/cases for any equation. These cases are ﬁltered by multiplying the entire expression with expressions of the form Arothi. c1h1Arothi + c2h2Arothi + c3h3Arothi + c4h4Arothi = h5Arothi (3) In the above equation, the signs of all expressions of the form hiArothj is known from the tournament conﬁguration. Therefore simply comparing the sign of the LHS and RHS for all possible values of irules out many cases. We now use multiple equations together in a bid to further ﬁlter the remaining cases. Consider the following equations without loss of generality. c1h1 + c2h2 + c3h3 + c4h4 = h5 (4) b1h2 + b2h3 + b3h4 + b4h5 = h6 (5) a1h1 + a2h2 + a3h3 + a4h4 = h6 (6) h5 can be eliminated from the ﬁrst two equations, leaving two expressions ofh6 in terms of h1,...h 4. The two sets of coefﬁcients of h1,...h 4 can be equated, and sign based arguments can eliminate a few more cases. Also, a set of 3 tuples can be constructed, with each item representing possible sign patterns for the 3 equations. Note that this set of 3 tuples does not contain entries which cannot apply simultaneously on the 3 equations. The above elimination step can be considered as a ﬁltration procedure given a size 6 tuple (h1,h2,h3,h4,h5,h6) by using 3 equations. We can perform a similar ﬁltration step given any size 6 tuple as well. Let the equations corresponding to (h1,h2,h3,h4,h5,h6) be E1,1,E1,2,E1,3. Similarly, consider the tuples(h1,h2,h3,h4,h5,h7), (h2,h3,h4,h5,h6,h7), (h1,h2,h3,h4,h6,h7) and their corresponding equations represented by Ei,j where iis the index of the tuple it corresponds to and jis the index of the equation. The above ﬁltration process can be performed on all the tuples, following which an additional ﬁltering step can be performed by using • E1,1 ≡E2,1 • E1,2 ≡E3,1 • E1,3 ≡E4,1 • E2,2 ≡E3,3 • E2,3 ≡E4,2 where the equivalence relation represents that the 2 equations are identical. Since identical equations must have identical sign patterns, the sign patterns not present in both sets of possible sign patterns can be ﬁltered out. For the 11-DRT cone, this leaves us with null set, proving that it is a forbidden conﬁguration. 14 Published as a conference paper at ICLR 2022 Proof of Theorem 9 Proof. The result follows the arguments in Forster & Simon (2006). We note that the arguments in Forster & Simon (2006) work only for non-zero sign matrices. By overloading T to denote the signed adjacency matrix of the corresponding tournament, we can consider the non-zero sign matrix G = T + diag(b) where b ∈{1,−1}n. By Gershgorin’s circle theorem and exploting the fact that any two rows of a doubly regular tournament are orthogonal, one can show that ρi(G)2 ≤ρi(T)2 + 2n−2 where ρi denotes the i-th largest singular value of the corresponding matrix. It then follows from Forster & Simon (2006) that if G has an representation in dim dimensions, then it must be the case that dim dim∑ i=1 (ρi(G))2 ≥n2. Noting that for a doubly regular cone T on nnodes, ρi(T).2 = n−1 ∀i, we get dim2 ·3(n−1) ≥n2 =>dim ≥ √n 3 (7) Thus to get a matrixM such that sign(M) = G, one needs at least√n 3 dimensions i.e., rank(M) ≥√n 3 . Now we show that this also is a lower bound for representing T. To see this, assume for the sake of contradiction that T has a representation in d dimension where d <√n 3 . Then, there exists H ∈Rd×n such that T = T[H]. The diagonal entries of T[H] are zero. Now introduce a small enough perturbation E to H and consider the matrix (H + E)TArotH. The entries of the perturbation matrix E ∈Rd×n can be chosen to be small enough such that the sign of the off-diagonal entries of HTArotH is same as that of (H + E)TArotH. However, the diagonal entries of (H + E)TArotH can get an arbitrary sign pattern, say b. Let G be the sign pattern matrix corresponding to (H + E)TArotH. By deﬁnition, G has a representation in dimension d< √n 3 . But this contradicts inequality (7). Thus, it must be the case that T has a representation in dimension at least √n 3 . Finally, setting n= 12d2 to the number of nodes in the doubly regular cone as given in the Theorem, we get that at least 2ddimensions are needed to embed such a tournament. The result follows. Proof of Theorem 10 Proof. The proof is the same arguments as the ﬁrst part of the proof of Theorem 9. Proof of Theorem 11 Proof. Given a tournament T, we ﬁrst show that we can start with an arbitrary transitive tournament and add enough rank 2 corrections to obtain a representation for T. Every addition of a rank 2 matrix will increase the representation dimension by at most 2. The result will follow then by noting that for the choice of the transitive tournament which minimizes the number of corrections needed, one needs at most 2(µ(T) + 1)representation dimension. Let T′ be an arbitrary transitive tournament which has a 2 dimensional representation and let M ∈Rn×n be the associated skew symmetric matrix that represents T′. W.l.o.g, assume that Mij >0 if and only if i<j . Deﬁne Ek(T) = {i: i<k, i< T k}. By deﬁnition, the feedback arc set Eσ(T) = ∪n k=1(k,Ek(T)) where σ = [1,...,n ] is the topological order corresponding to the transitive tournament T′. We start with k= nand correct the feedback arc errors arising from Ek iteratively. Deﬁne ∆n = √ max i<n (Min + ϵ) for some small enough ϵ> 0. Let u ∈Rn be such that ui = ∆ ∀i∈En and ui = 0 otherwise. Let v ∈Rn be such that vn = −∆,vi = 0 ∀i̸= n. It is easy to verify that M+uvT −vuT represents a tournament that has the feedback arc set∪n−1 k=1 (k,Ek(T)) i.e., the errors in En have been corrected. The cost of correcting the error is adding a rank 2 skew 15 Published as a conference paper at ICLR 2022 symmetric matrix which increases the representation dimension by at most 2. One can repeat the same procedure for n−1,n −2,... until all errors are corrected. The upper bound in the theorem follows noting that the above argument works for any transitive tournament T′and so we can start with the one which has the least number of nodes involved in the feedback arc set to minimize the number of extra dimensions needed to represent T. Proof of Theorem 12 Proof. From Theorem 11, T can be represented using at most 2(µ(T) + 1)dimensions. By construc- tion any representation of T must also represent G. The result follows. A.0.2 A DDITIONAL FILTRATION FOR VERIFYING 10 N ODE FORBIDDEN CONFIGURATION T =   0 1 1 1 −1 −1 −1 −1 −1 1 −1 0 1 −1 1 1 −1 −1 1 −1 −1 −1 0 1 1 −1 1 −1 −1 −1 −1 1 −1 0 −1 1 1 −1 −1 −1 1 −1 −1 1 0 1 −1 −1 1 −1 1 −1 1 −1 −1 0 1 −1 1 −1 1 1 −1 −1 1 −1 0 −1 1 −1 1 1 1 1 1 1 1 0 1 −1 1 −1 1 1 −1 −1 −1 −1 0 −1 −1 1 1 1 1 1 1 1 1 0   Due to the lesser number of constraints present when dealing with 10 nodes instead of 12, an additional ﬁltration step is required. The entire procedure described above can be considered as an elimination procedure given a tuple T1 = (h1,...h 7). Also consider that in the ﬁnal ﬁltration above, a set of size 4 tuples is created. These size 4 tuples will represent the possible sign patterns for E1,1,E1,2,E1,3,E2,2. If a similar procedure is carried out on T2 = (h2,...h 8), another set of 4 tuples will be generated. The ﬁltration in this step is based on the fact that the equations E1,3,E2,2 corresponding to T1 are the same as E1,1,E1,2 for T2. The sign patterns which are not present in both sets of tuples are ﬁltered out. This type of relationship can be obtained between T2 and T3 = (h3,...h 8,h0) as well, and in general between any Ti and Tj such that Tj can be obtained from Ti by removing the ﬁrst entry and adding an entry to the end. Note that the tuples must have unique entries. Therefore a ’circular elimination’ procedure can be followed, where the Ti tuples considered are size 7 subsets of the circular permutations of h1,...h 8. By following this procedure, all the sign patterns remaining can be eliminated. Using size 7 circular permutations of h1,...h 10 in the last step does not eliminate all the cases, we believe this is due to nodes 1 to 8 forming a 7-DRT cone, which leads to many eliminations. The code for the ﬁltration can be found here. 16	Arun Rajkumar, Vishnu Veerathu, Abdul Bakey Mir	Accept (Poster)	2,022	{"id": "zzk231Ms1Ih", "original": "_vWu7gLdTrwb", "cdate": 1632875730339, "pdate": 1643407560000, "odate": 1633539600000, "mdate": null, "tcdate": 1632875730339, "tmdate": 1676330471169, "ddate": null, "number": 4095, "content": {"title": "A Theory of Tournament Representations", "authorids": ["~Arun_Rajkumar4", "~Vishnu_Veerathu1", "cs20d400@smail.iitm.ac.in"], "authors": ["Arun Rajkumar", "Vishnu Veerathu", "Abdul Bakey Mir"], "keywords": ["tournament", "skew-symmetric", "pairwise ranking"], "abstract": "Real-world tournaments are almost always intransitive. Recent works have noted that parametric models which assume $d$ dimensional node representations can effectively model intransitive tournaments. However, nothing is known about the structure of the class of tournaments that arise out of any fixed $d$ dimensional representations. In this work, we develop a novel theory for understanding parametric tournament representations. Our first contribution is to structurally characterize the class of tournaments that arise out of $d$ dimensional representations. We do this by showing that these tournament classes have forbidden configurations that must necessarily be a union of flip classes, a novel way to partition the set of all tournaments. We further characterize rank $2$ tournaments completely by showing that the associated forbidden flip class contains just $2$ tournaments. Specifically, we show that the rank $2$ tournaments are equivalent to locally transitive tournaments. This insight allows us to show that the minimum feedback arc set problem on this tournament class can be solved using the standard Quicksort procedure. We also exhibit specific forbidden configurations for rank $4$ tournaments. For a general rank $d$ tournament class, we show that the flip class associated with a coned-doubly regular tournament of size $\\mathcal{O}(\\sqrt{d})$ must be a forbidden configuration. To answer a dual question, using a celebrated result of Froster, we show a lower bound of $\\Theta(\\sqrt{n})$ on the minimum dimension needed to represent all tournaments on $n$ nodes. For any given tournament, we show a novel upper bound on the smallest representation dimension that depends on the least size of the number of unique nodes in any feedback arc set of the flip class associated with a tournament. We show how our results also shed light on the upper bound of sign-rank of matrices. ", "one-sentence_summary": "We develop a theory to understand tournament representations i.e. structurally characterise when a tournament graph can be represented in lower dimensions using a skew symmetric matrix. ", "code_of_ethics": "", "submission_guidelines": "", "resubmission": "", "student_author": "", "serve_as_reviewer": "", "paperhash": "rajkumar\|a_theory_of_tournament_representations", "pdf": "/pdf/a7853d8c301f8a37bc858f4c428d73862dabff26.pdf", "_bibtex": "@inproceedings{\nrajkumar2022a,\ntitle={A Theory of Tournament Representations},\nauthor={Arun Rajkumar and Vishnu Veerathu and Abdul Bakey Mir},\nbooktitle={International Conference on Learning Representations},\nyear={2022},\nurl={https://openreview.net/forum?id=zzk231Ms1Ih}\n}", "venue": "ICLR 2022 Poster", "venueid": "ICLR.cc/2022/Conference"}, "forum": "zzk231Ms1Ih", "referent": null, "invitation": "ICLR.cc/2022/Conference/-/Blind_Submission", "replyto": null, "readers": ["everyone"], "nonreaders": [], "signatures": ["ICLR.cc/2022/Conference"], "writers": ["ICLR.cc/2022/Conference"]}	[Main Review]: I think this paper has a solid theoretical foundation of an interesting problem on tournaments. My main concern is if this conference is the right venue since I fail to see the connection to "learning." The authors briefly address applications in Section 7.2, but I think this section should be expanded. Some of the proofs can be moved to the appendices to make space if need be. Minor errors: “Recent works have noted that parametric models which assume d dimensional node representations can effectively model intransitive tournaments” are there any citations for this statement? You might want to list them in the introduction where something similar is also mentioned. No “.” after 2) point at the end of Page 1. In the definition of skew symmetric, it would be easy to add A^T = - A The name of the Lemma is Farkas’ Lemma, and not Farka’s lemma. In the citation of Charbit et al, the “np” should be capitalized.	ICLR.cc/2022/Conference/Paper4095/Reviewer_MMVk	null	4	{"id": "pUE59TP-ax", "original": null, "cdate": 1635978344716, "pdate": null, "odate": null, "mdate": null, "tcdate": 1635978344716, "tmdate": 1635978344716, "ddate": null, "number": 4, "content": {"summary_of_the_paper": "A tournament is made by choosing a direction for each of the edges in a complete graph. A tournament can be induced of by skew symmetric matrices M where entries M_{ij} > 0 if and only if (i,j) is an edge. A tournament on n edges can be represented by a set of d-dimensional vectores {h_1, \u2026 , h_n} if (h_j)^T A h_i is not zero iff (i,j) is an edge (and A is an appropriate matrix).\n\nThe authors address two questions:\n\n1) What structurally characterizes the class of tournaments that can be represented in d dimensions?\n\n2) Given a tournament T on n nodes, what is the minimum dimension d needed to represent it?.\n\nThe first question is answered by considering structures the authors called forbidden. The authors provide a characterization of these forbidden structures as a union of certain equivalence classes. \n\nThey answer the latter question in part by providing bounds on what said dimension d should be. \n\n", "main_review": "I think this paper has a solid theoretical foundation of an interesting problem on tournaments.\n\nMy main concern is if this conference is the right venue since I fail to see the connection to \"learning.\" The authors briefly address applications in Section 7.2, but I think this section should be expanded. Some of the proofs can be moved to the appendices to make space if need be. \n\nMinor errors:\n\n\u201cRecent works have noted that parametric models which assume d dimensional node representations can effectively model intransitive tournaments\u201d are there any citations for this statement? You might want to list them in the introduction where something similar is also mentioned. \n\n\nNo \u201c.\u201d after 2) point at the end of Page 1.\n\nIn the definition of skew symmetric, it would be easy to add A^T = - A\n\nThe name of the Lemma is Farkas\u2019 Lemma, and not Farka\u2019s lemma.\n\nIn the citation of Charbit et al, the \u201cnp\u201d should be capitalized. ", "summary_of_the_review": "Overall, I think the paper should be accepted provided that the authors make more of an effort to relate their results to learning. ", "correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "1: The contributions are neither significant nor novel.", "flag_for_ethics_review": ["NO."], "recommendation": "8: accept, good paper", "confidence": "4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work."}, "forum": "zzk231Ms1Ih", "referent": null, "invitation": "ICLR.cc/2022/Conference/Paper4095/-/Official_Review", "replyto": "zzk231Ms1Ih", "readers": ["everyone"], "nonreaders": [], "signatures": ["ICLR.cc/2022/Conference/Paper4095/Reviewer_MMVk"], "writers": ["ICLR.cc/2022/Conference", "ICLR.cc/2022/Conference/Paper4095/Reviewer_MMVk"]}	{ "criticism": 2, "example": 1, "importance_and_relevance": 1, "materials_and_methods": 1, "praise": 1, "presentation_and_reporting": 6, "results_and_discussion": 0, "suggestion_and_solution": 5, "total": 9 }	1.888889	1.63637	0.252519	1.896583	0.180045	0.007695	0.222222	0.111111	0.111111	0.111111	0.111111	0.666667	0	0.555556	{ "criticism": 0.2222222222222222, "example": 0.1111111111111111, "importance_and_relevance": 0.1111111111111111, "materials_and_methods": 0.1111111111111111, "praise": 0.1111111111111111, "presentation_and_reporting": 0.6666666666666666, "results_and_discussion": 0, "suggestion_and_solution": 0.5555555555555556 }	1.888889	iclr2022	openreview	0	0			0	null
zzk231Ms1Ih	A Theory of Tournament Representations	Real-world tournaments are almost always intransitive. Recent works have noted that parametric models which assume $d$ dimensional node representations can effectively model intransitive tournaments. However, nothing is known about the structure of the class of tournaments that arise out of any fixed $d$ dimensional representations. In this work, we develop a novel theory for understanding parametric tournament representations. Our first contribution is to structurally characterize the class of tournaments that arise out of $d$ dimensional representations. We do this by showing that these tournament classes have forbidden configurations that must necessarily be a union of flip classes, a novel way to partition the set of all tournaments. We further characterize rank $2$ tournaments completely by showing that the associated forbidden flip class contains just $2$ tournaments. Specifically, we show that the rank $2$ tournaments are equivalent to locally transitive tournaments. This insight allows us to show that the minimum feedback arc set problem on this tournament class can be solved using the standard Quicksort procedure. We also exhibit specific forbidden configurations for rank $4$ tournaments. For a general rank $d$ tournament class, we show that the flip class associated with a coned-doubly regular tournament of size $\mathcal{O}(\sqrt{d})$ must be a forbidden configuration. To answer a dual question, using a celebrated result of Froster, we show a lower bound of $\Theta(\sqrt{n})$ on the minimum dimension needed to represent all tournaments on $n$ nodes. For any given tournament, we show a novel upper bound on the smallest representation dimension that depends on the least size of the number of unique nodes in any feedback arc set of the flip class associated with a tournament. We show how our results also shed light on the upper bound of sign-rank of matrices.	Published as a conference paper at ICLR 2022 A THEORY OF TOURNAMENT REPRESENTATIONS Arun Rajkumar Indian Institute of Technology RBCDSAI, IITM Abdul Bakey Mir Indian Institute of Technology Vishnu Veerathu Cohesity Inc ABSTRACT Real world tournaments are almost always intransitive. Recent works have noted that parametric models which assume d dimensional node representations can effectively model intransitive tournaments (Rajkumar & Agarwal (2016)). How- ever, nothing is known about the structure of the class of tournaments that arise out of any ﬁxed ddimensional representations. In this work, we develop a novel theory for understanding parametric tournament representations. Our ﬁrst contri- bution is to structurally characterize the class of tournaments that arise out of d dimensional representations. We do this by showing that these tournament classes have forbidden conﬁgurations which must necessarily be union of ﬂip classes, a novel way to partition the set of all tournaments. We further characterize rank 2 tournaments completely by showing that the associated forbidden ﬂip class con- tains just 2 tournaments. Speciﬁcally, we show that the rank 2 tournaments are equivalent to locally-transitive tournaments. This insight allows us to show that the minimum feedback arc set problem on this tournament class can be solved using the standard Quicksort procedure. For a general rank dtournament class, we show that the ﬂip class associated with a coned-doubly regular tournament of size O( √ d) must be a forbidden conﬁguration. To answer a dual question, using a celebrated result of Forster & Simon (2006), we show a lower bound of Ω(√n) on the minimum dimension needed to represent all tournaments on nnodes. For any given tournament, we show a novel upper bound on the smallest representation dimension that depends on the least size of the number of unique nodes in any feedback arc set of the ﬂip class associated with a tournament. We show how our results also shed light on upper bound of sign-rank of matrices. 1 I NTRODUCTION In this work, we lay the the foundations for a theory of tournament representations. A tournament is a complete directed graph and arises naturally in several applications including ranking from pairwise preferences, sports modeling, social choice, etc. We say that a tournament T on nnodes can be represented in ddimensions if there exists a skew symmetric matrix M ∈Rn×n of rank dsuch that a directed edge from ito j is present in T if and only if Mij > 0. Real world tournaments are almost always intransitive (Tversky (1969); Klimenko (2015)) and it is not known what type of tournaments can be represented in how many dimensions. This is important to understand because of the following reason: As a modeler of preference relations using tournaments, it is often more natural to have structural domain knowledge such as ‘The tournaments under consideration do not have long cycles’as opposed to algebraic domain knowledge such as ‘The rank of the skew symmetric matrix associated with the tournaments of interest is at most k’. However, algorithms that learn rankings from pairwise comparison data typically need as input the algebraic quantity - the rank of the skew symmetric matrices associated with tournaments or equivalently the dimension where they are represented (Rajkumar & Agarwal (2016)). To bridge the gap between the structural and the algebraic world, we ask and answer two fundamental questions regarding the representations of tournaments. 1) What structurally characterizes the class of tournaments that can be represented in ddimensions? 2) Given a tournament T on nnodes, what is the minimum dimension dneeded to represent it?. 1 Published as a conference paper at ICLR 2022 … FLIP CLASS 1 FLIP CLASS 2 FLIP CLASS k Locally Transitive Transitive R- cone Forbidden2d-Representable ` FLIP CLASS 1 FLIP CLASS 2 ForbiddenRepresentable Figure 1: (Left) Partitions of the set of all tournaments on nnodes using ﬂip classes. Every shaded region is a ﬂip class partition and every circle indicates a tournament. The ﬂip class that contains the transitive tournament (Flip class 1) is precisely the set of all locally transitive tournaments. This is also the set of all tournaments that can be represented in 2 dimensions (Section 5). Every ﬂip class contains a canonical representative termed the R-cone (Section 4), indicated using the larger circle inside each ﬂip class. The tournaments that cannot be represented using ddimensions appear as union of Forbidden Flip classes (Flip class kin Figure) (Section 4). (Right) Explicit ﬂip class partition of the 4 possible non-isomorphic tournaments on 4 nodes. Tournaments in ﬂip class 1 can be represented using 2 dimensions whereas tournaments in ﬂip class 2 cannot (see Section 5). . We answer the ﬁrst question by investigating the intricate structure of the rank dtournament class via the notion of forbidden conﬁgurations. Speciﬁcally, we show that the set of forbidden conﬁg- uration for the rank dtournament class must necessarily be a union of ﬂip classes, a novel way to partition the set of all tournaments into equivalence classes. We explicitly characterize the forbidden conﬁgurations for the rank 2 tournament class and exhibit a forbidden conﬁguration for the general rank dtournament class. Speciﬁcally, we show that the rank 2 tournaments are equivalent to locally transitive tournaments, a previously studied class of tournaments (Cohen et al. (2004)). Our results throw light on the connections between transitive and locally transitive tournaments and also lets us develop a classic Quicksort based algorithm to solve that the minimum feedback arc set problem on rank 2 tournaments with O(n2) time complexity. Our results for the general rank dtournament class have connections to the classic long standing Hadamard conjecture and we discuss this as well. Figure 1 gives a glimpse of some of the main results. We answer the second question by proving lower and upper bounds on the smallest dimension needed to represent a tournament on nnodes. We exhibit a lower bound of Ω(√n) using a variation of the celebrated dimension complexity result of Forster & Simon (2006) for sign matrices. To show upper bounds, we introduce a novel parameter associated to a tournament called the Flip Feedback Node set of a Tournament. This quantity depends on the least number of unique nodes in any feedback arc set of an associated tournament class for the tournament of interest and upper bounds linearly the representation dimension of any tournament. We show how our results can be used to provide upper bounds on the classic notion of sign-rank of a matrix. Previously known upper bounds for sign rank depended on the VC dimension of the associated binary function class (Alon et al. (2016)). Our upper bounds on the other hand have a graph theoretic ﬂavour. Organization of the Paper: We discuss brieﬂy in Section 2 the foundational works this paper builds upon. We introduce necessary preliminaries in Section 3. The answer to the ﬁrst question about the structural characterization of ddimensional tournament classes span Sections 4, 5 and 6. We devote Section 7 to answer the second question about the number of dimensions needed to represent a tournament. Section 8.1 explores connections of our results to upper bounds on the sign rank of a sign matrix. Finally, we conclude in Section 10. All proofs are defered to the Appendix. 2 R ELATED WORK The work in this paper builds on several pieces of work across different domains. We summarize below the most important related works under different categories: 2 Published as a conference paper at ICLR 2022 Intransitive Pairwise Preference Models: One of the main reasons to study representations of tournaments is to model pairwise preferences. Parametric pairwise preference models that can model intransitivity have gained recent interest. Rajkumar & Agarwal (2016) develop a low rank pairwise ranking model that can model intransitivity. However, their study and results were restricted to just the transitive tournaments in these classes. A generalization of the classical Bradley-Terry-Luce model (Bradley & Terry (1952), Luce (2012)) was studied in Causeur & Husson (2005). However, no structural characterization is known. Same holds for the more recent models studied in Chen & Joachims (2016), Makhijani & Ugander (2019) and Bower & Balzano (2020). Flip Classes: The notion of ﬂip classes, a novel way to partition the set of all tournaments on n nodes, was ﬁrst introduced in Fisher & Ryan (1995). The goal however was completely different and was on studying equilibrium on certain generalized rock-paper-scissors games on tournaments. Interestingly, and perhaps surprisingly, the notion of ﬂip classes turn out to be fundamental to our study of understanding forbidden conﬁgurations of ddimensional tournament classes. Dimension Complexity and Sign Rank: Dimension complexity and sign rank of sign pattern matrices were studied in Forster (2002). These results have found signiﬁcant applications in learning theory and lower bounds in computational complexity (Forster & Simon (2006)). More recently, Alon et al. (2016) study the sign rank for function classes with ﬁxed Vapnik-Chervonenkis (VC) dimension and show upper bounds. Our upper bounds however depend on certain graph theoretic properties. 3 P RELIMINARIES Tournaments: A tournament T is a complete directed graph i.e., a graph on nnodes where for every pair of nodes (i,j) either an edge is oriented from ito jor jto i. The number of nodes nin T will be usually clear from the context or will be explicitly speciﬁed. For nodes iand jin T, we say i≻T j if there is a directed edge from ito j. Given a node i, we deﬁne the out and in neighbours of ias T+ i = {j : i ≻T j}and T− i = {j : j ≻T i}respectively. Given a set of nodes S, we denote by T(S) the induced sub-tournament of T on the nodes in S. Feedback Arc Set and Pairwise Disagreement Error: Given a permutation σ on n nodes, the feedback arc set of σw.r.t the tournament T is deﬁned as Eσ(T) = {(i,j) : σ(i) >σ(j),i> T j}. The pairwise diagreement error of σ w.r.t T is given by \|Eσ(T)\| ( n 2) . It is known that ﬁnding the σ that minimizes the pairwise disagreement error w.r.t a general tournamentT is a NP-hard problem (Charbit et al. (2007)). Skew Symmetric Tournament Classes: A square matrix M ∈Rn×n is skew symmetric if Mij = −Mji∀i,j. In this paper, whenever we refer to a skew symmetric matrix M ∈Rn×n, we always assume Mij ̸= 0 ∀i̸= jand Mii = 0 ∀i. Given such an M, we denote by T{M}the tournament on nnodes induced by M where i≻T j ⇐⇒Mij >0. We refer to class of tournaments induced by rank dskew symmetric matrices as the rank dtournament class. Forbidden Conﬁgurations: A tournament class Tis a collection of tournaments. Tis said to forbid a tournament T if no tournament in T has a sub-tournament that is isomorphic to T. We call T a forbidden conﬁguration for T if T forbids T but does not forbid any sub-tournament of T. For example, the class of all acyclic/transitive tournaments has the 3-cycle as a forbidden conﬁguration i.e, the tournament T on three nodes i,j,k where i≻T j ≻T k≻T i. Representations and Tournaments: The matrix Arot ∈Rd×d is deﬁned for every even dand is a block diagonal matrix which consists ofd/2 blocks of [0−1; 1 0]. This is the canonical representation of a non-degenerate skew symmetric matrix i.e., any non-singular skew symmetric matrix can be brought to this form by a suitable basis transformation. Given a set of vectors in H = {h1,..., hn} where each hi ∈Rd for some even d, we refer to the set H as a tournament inducing representation if hT i Arothj ̸= 0 for all i ̸= j. Furthermore, we refer to the tournament induced by H as T[H] where a directed edge exits from node ito jif and only if hT i Arothj >0. It is easy to verify that hTAroth = 0 for any h ∈Rd. Any skew symmetric matrix M ∈Rn×n of rank dcan be written as M = HTArotH for some H ∈Rd×n. It follows that if M = HTArotH, then T{M}= T[H]. Positive Spans: A set of vectors H = {h1,..., hn}where each hi ∈Rd is said to positively span Rd if for any w ∈Rd, there exists non-negative constants c1,...,c n ≥0 such that ∑ icihi = w. If 3 Published as a conference paper at ICLR 2022 H positively spans Rd, then there does not exist a w ∈Rd such that wThi >0 ∀i. This is a easy consequence of Farkas’ Lemma. Remark on Notation: We reiterate that we use T(·),T{·}and T[·] to mean different objects - the tournament induced by a subset of nodes, the tournament induced by a skew symmetric matrix and the tournament induced by a representation of a set of vectors respectively. These will be usually clear from the context. 4 F LIP CLASSES , FORBIDDEN CONFIGURATIONS AND POSITIVE SPANS The main purpose of this section is understand the space of forbidden conﬁgurations of rank d tournaments. The main result of this section shows that the forbidden conﬁgurations for rank d tournament classes occur as union of certain carefully deﬁned equivalence classes of non-isomorphic tournaments. Towards this, we deﬁne the notion of ﬂip classes which was introduced ﬁrst in (Fisher & Ryan, 1995) although in a different context: Deﬁnition 1. Given a tournamentT on nnodes and a setS ⊆[n], deﬁne φS(T) to be the tournament obtained from T by reversing the orientation of all edges (i,j) such that i∈S,j ∈¯S. In other words, φS(T) is obtained from T by reversing the edges across the cut (S, ¯S) = {(i,j) : i∈S,j ∈¯S}. Deﬁnition 2. A class of tournaments on nnodes is called cut-equivalent if for every pair of tourna- ments T,T′in the class, there exists a S ⊆[n] such that T′is isomorphic to φS(T) It is easy to show that the set of cut-equivalent tournaments form an equivalence relation over the set of all tournaments on nnodes Fisher & Ryan (1995). The corresponding equivalence classes are called a ﬂip classes. We denote by F(T) the ﬂip class of T i.e., the equivalence class of all cut-equivalent tournaments to T. In the following theorem we show the fundamental relation between ﬂip classes and forbidden conﬁgurations. Theorem 1. Let T, a tournament on knodes, be a forbidden conﬁguration for some rank dtourna- ment class. Then every tournament in the ﬂip class of T is also a forbidden conﬁguration for the rank dtournament class. Corollary 1. (Structure of Forbidden conﬁgurations) Forbidden conﬁgurations of rankdtourna- ment classes are unions of ﬂip classes. Thus, to characterize the forbidden conﬁguration of rank dtournament classes, we need to understand the ﬂip classes. We begin with the following simple but useful deﬁnition. Deﬁnition 3. A tournament T is called a R-cone if there exists a node i∗such that • i∗≻T jfor all j ̸= i∗( j ≻T i∗for all j ̸= i∗) • T(T+ i∗) ( T(T− i∗) ) is isomorphic to the tournament R. R-cones are essentially the tournament R along with an additional node that either beats or loses to nodes in R. R-cones are useful as they be viewed in some sense as canonical tournaments in ﬂip classes. This is justiﬁed because of the following observation. Proposition 1. Every ﬂip class contains an R-cone for some tournament R. The above observation says that to identify the forbidden conﬁgurations for a given tournament class, it sufﬁces to identify all forbidden R-cones. Then by Corollary 1, the associated ﬂip classes will be the set of all forbidden conﬁgurations. However, it does not throw light on what property the tournament R must satisfy. The following lemma establishes this. Lemma 2. Let R be a tournament with the property that if R = R[H] for some representation H = {h1,..., hn} ∈Rd then H positively spans Rd. Then rank d tournament class forbids R-cones. The above lemma is extremely helpful in the sense that it reduces the study of forbidden conﬁgurations to the study of ﬁnding tournaments such that any representation that induces it must necessarily 4 Published as a conference paper at ICLR 2022 positively span the entire Euclidean space. Note that there may be several representations which positively span the entire space. This does not mean their the associated coned tournaments are forbidden conﬁgurations. Instead, we start with an R cone and conclude it is forbidden if every representation that induces R necessarily positively spans the entire space. It is a non-trivial problem to identify such tournaments R for an arbitrary dimension d. In the following sections, we explicitly identify the only forbidden ﬂip class for rank 2 tournaments, one forbidden ﬂip class for rank 4 and then (a potentially weaker) forbidden ﬂip class for the general rank dcase. 5 R ANK 2 TOURNAMENTS ⇐⇒LOCALLY TRANSITIVE TOURNAMENTS The goal of this section is to characterize the forbidden conﬁgurations of rank 2 tournaments. Thanks to Lemma 2, this reduces to the problem of identifying a tournament whose representation necessarily span the entire space. The following lemma exhibits this tournament. Lemma 3. Let H = {h1,h2,h3}∈ R2 be two dimensional representation of 3 nodes which induce a 3 cycle tournament. Then the set H positively spans R2. Furthermore, the 3 cycle is the only such tournament on 3 nodes. The above lemma immediately implies that a coned 3-cycle is the only forbidden conﬁguration for rank 2 matrices. This is indeed true. However, for the purposes of generalizing our result (which as we will see will be useful when discussing the higher dimension case), we will view the 3-cycle as a special case of a doubly regular tournament (deﬁned next). Deﬁnition 4. A tournament T on nnodes is said to be n-doubly regular if \|T+ i \|= \|T+ j \|for all i,j and \|T+ i ∩T+ j \|= kfor some ﬁxed kfor all i̸= j Trivially the 3-cycle is the only 3-doubly regular tournament. The following lemma establishes that the ﬂip class of a coned 3 doubly regular tournament only contains itself. Proposition 2. The ﬂip class of 3-doubly-regular-cone does not contain any other tournament. Theorem 4. 3-doubly-regular-cone is the only forbidden conﬁguration for the rank 2 tournament class. We have thus far established that any rank2 tournament on nnodes forbids the 3-doubly-regular-cone. The advantage of this result is that we can go one step further and explicitly characterize the rank 2 tournament class. To do this, we need the deﬁnition of a previously studied tournament class. Deﬁnition 5. A tournament T is called locally transitive if for every node iin T, both T(T+ i ) and T(T− i ) are transitive tournaments Before seeing why locally transitive tournaments are relevant to our study, we ﬁrst show that they are intimately connected to transitive tournaments via the following characterization. Theorem 5. (Connection between Transitive and Locally Transitive Tournaments) The set of all non isomorphic locally transitive tournaments on n nodes is equivalent to the ﬂip class of the transitive tournament on nnodes. We will see next that this key result allows us to immediately characterize the rank 2 tournament class. We will see later (Section 7) that this result is also crucial in determining an upper bound on the dimension needed to represent a given tournament. Theorem 6. (Characterization of rank 2 tournaments) A tournament T on n nodes is locally transitive if and only if there exists a skew symmetric matric M ∈Rn×n with rank(M) = 2 such that the T{M}= T. It is perhaps surprising that a purely structural description of a tournament class namely that of local transitivity turns out to be exactly equivalent to the rank 2 tournament class. To the best of our knowledge, this characterization appears novel and hasn’t been previously noticed. One of the interesting consequence of the above characterization is that the minimum feedback arc set problem on rank 2 tournaments can be solved using a standard quick sort procedure. This is formalized below. 5 Published as a conference paper at ICLR 2022 Theorem 7. (Minimum Feedback Arc Set is Poly-time Solvable for Rank 2 tournaments) Let T be a locally transitive tournament on nnodes and let σ1 be the permutation returned by running a standard quick sort algorithm choosing 1 as the initial pivot node and where the outcome of comparison between items iand j is iif and only if i≻T j. Let σk be obtained from σ1 by k−1 clockwise cyclic shifts for k ∈[n]. Let Ek be the feedback arc set of σk w.r.tT. Then mink\|Ek\| achieves the minimum size of the feedback arc set for T. Proof. (Sketch) The proof involves two steps: ﬁrst arguing that by ﬁxing any pivot, quick sort would return a ranking that is a cyclic shift of σ1. The second step involves inductively arguing that one of the cyclic shift must necessarily minimize the feedback arc set. 5.1 R ANK 4 TOURNAMENTS We now turn to rank 4 tournaments. We could have directly considered rank dtournaments, but it turns out that what we can show a slightly stronger result for rank 4 tournaments than the general case and so we focus on them separately. While it is arguably simple in the rank 2 case to identify the tournament that necessitates the positive spanning property, it is not immediately clear in the rank 4 case. A ﬁrst guess would be to consider the regular tournaments (as the 3 cycle for rank 2 is also a regular tournament) on 5 nodes or 7 nodes. However, these turn out to be insufﬁcient as one can construct counter examples of regular tournaments on up to 7 nodes with representations that don’t span the entire R4. In fact, as we had deﬁned earlier, the right way to generalize to higher dimension turns out to be using doubly regular tournaments. Theorem 8. The 11-doubly-regular-cone is a forbidden conﬁguration for rank 4 tournament class. Note that while for the rank 2 case, we were able to prove that the only forbidden ﬂip class is the one that contains a coned 3 cycle, we have not shown that the only forbidden conﬁguration for rank 4 class is the 11 doubly regular cone. In fact, we believe that the smallest forbidden tournament for rank 4 class is the 7- doubly regular cone. However, we haven’t been able to prove this. This appears to be a non-trivial problem. From our simulation experiments, we observe that the only ﬂip class that could be forbidden on 8 nodes is the one that contains the 7-doubly regular cone. In particular, we were able to produce examples of representations for all other ﬂip classes on 8 nodes. However, this does not imply that the ones where we could not produce a forbidden conﬁguration is in fact forbidden. Unfortunately, it seems tricky to prove this and we don’t have a way to show this at this point. On the other hand, as we will see in the next section, the result in Theorem 8 is still stronger than the result for the general rank dtournaments. Having discussed rank 2 and rank 4 cases separately, we next turn our attention to the general rank d tournament class. 6 R ANK d TOURNAMENTS AND THE HADAMARD CONJECTURE From the understanding of rank 2 and rank 4 tournament classes in the previous sections, and noting that the corresponding forbidden conﬁgurations are intimately related to doubly regular tournaments, it is tempting to conjecture that this is true in general. Conjecture 1. Rank 2dtournament class forbids 4d−1-doubly-regular-cones. Ideally, the conjecture above must have a qualiﬁer ‘if they exist’ for the 4d−1 doubly regular cone. This is because of the equivalence between doubly regular tournaments on 4d−1 nodes and Hadamard matrices in {+1,−1}4d×4d (Reid & Brown (1972)). A matrix H ∈{+1,−1}n×n is called Hadamard if H′H = nI where I is the identity matrix. It is known that there is a bijection between skew Hadamard matrices and doubly regular tournaments Reid & Brown (1972). A long standing unsolved conjecture about Hadamard matrices is the following: Conjecture 2. (Hadamard) There exists a Hadamard matrix of order4dfor every d> 0. If Conjecture 1 were true, then it would imply the existence of 4d−1-doubly regular tournament for every dand thus would imply the Hadamard conjecture is true. In fact, Conjecture 1 being true would say more which we state below: Conjecture 3. There exists a skew symmetric Hadamard matrix of order4dfor every d> 0. 6 Published as a conference paper at ICLR 2022 The main result of this section is a weaker form of the conjecture: Theorem 9. Rank 2(d−1) tournament class forbids 12d2 −1-doubly-regular-cones if they exist. 7 H OW MANY DIMENSIONS ARE NEEDED TO REPRESENT A TOURNAMENT ? The previous sections considered a speciﬁc rank dtournament class and tried to characterize them using forbidden conﬁgurations. In this section, We turn to the dual question of understanding the minimum number of dimensions needed to embed a tournament. We start by not considering a single tournament T but the set of all tournaments on nnodes. We show below a general result which provides a lower bound on the minimum dimension needed to embed any tournament on nnodes. Theorem 10. (Lower Bound on minimum representation dimension) Let T be a tournament on n nodes. Then there existsH = {h1,...,h n}∈ Rdsuch that T = T[H] only if d∑n i=1(ρi(T+I))2 ≥ n2. Furthermore, let dbe the minimum dimension needed to embed every tournament on nnodes. Then, d≥√n. The result follows the arguments in the celebrated work of Forster & Simon (2006). As Alon et al. (2016) point out, Forster’s technique cannot be stretched further in obvious ways to get upper bounds. The above theorem tells us that in the worst case at leastΩ(√n) dimensions are necessary to represent all tournaments on nnodes. In fact if Conjecture 1 were true, the minimum dimension would be Ω(n). However, in practice one might not encounter tournaments with such extremal/worst-case properties. Remark: The lower bound for the representation dimension of random tournaments (where the orientation of each edge is determined by an unbiased Bernoulli coin toss) will depend on the singular values of random tournaments. However, we don’t expect the representation dimension to be of independent of n, the number of nodes. Loosely speaking, a doubly regular tournament is "like" a random tournament (the associated Hadamard Tournament has been used in deterministic perturbation schemes as alternatives for random sign matrices (Bhatnagar et al. (2003))). On the other hand, the most interesting real-world tournaments might be characterized by constant sized node representations and hence may be structurally much more constrained than random tournaments. Typically, a smaller number of dimensions might be enough to represent tournaments of practical interest. Our goal below is to upper bound on the number of dimensions needed to embed a given tournament T. Recall that Eσ(T) denotes the feedback arc set of a permutation σ w.r.t a tournament T. We deﬁne the number of nodes involved in the feedback arc set as follows: θ(σ,T) = \|{i: ∃j : σ(i) >σ(j),(i,j) ∈Eσ(T)}\|. We next deﬁne a crucial quantity µ(T) which we term as the Flip Feedback Node set size. This quantity will determine an upper bound on the dimension where a tournament can be represented: Deﬁnition 6. Given a tournament T, deﬁne the Flip Feedback Node Set Size as follows: µ(T) = min σ min T′∈F(T) θ(σ,T′) In words, given a tournament T, the quantity µ(T) captures the minimum number of nodes involved in any feedback arc set among all tournaments in the ﬂip class of T. For instance if T is a locally transitive tournament then µ(T) would be 0 - as T is necessarily in the ﬂip class of a transitive tournament and so the Eσ corresponding to the topological ordering of the transitive tournament would have an empty feedback arc set. As another example, consider T to be the coned 3-cycle. The ﬂip class of this tournament contains only itself and the best permutation will have one edge in the feedback arc set. Thus µ(T) = 1. In general, it is trivially true that µ(T) is upper bounded by n, the number of nodes. Howeverµ(T) could be much smaller than ndepending on T. The main result of this section is the theorem below that shows that µ(T) gives an upper bound on the number of dimension needed to represent any tournament. Theorem 11. (Upper Bound on minimum representation dimension) For any tournamentT, there exists H = {h1,..., hn}∈ R2(µ(T)+1) such that T = T[H]. 7 Published as a conference paper at ICLR 2022 1 2 3 4 5 6 ` Figure 2: A simple tournament to illustrate that the quantity µ(T) need not be same as the size of the minimum feedback arc set. All edges go from left to right except the ones in red. See Remark 2 in Section 7 for details. Remark 1: The above theorem says that one can always obtain an representation H in dimension d= O(µ(T)) that induces T. The bound gets tighter for tournaments with smaller feedback arc sets, which is what one might typically expect in practice. Note that even for some tournaments that may have a large feedback arc set, the associated ﬂip class might contain a tournament with a smaller feedback arc set. Remark 2: We note that in general µ(T) is not necessarily the cardinality of the minimum feedback arc set among all tournaments in the ﬂip class of T. Instead µ(T) captures the cardinality of the set of nodes involved in any feedback arc set. To see why these two could be different, consider the tournament in Figure 2. Here, σa = [6 1 2 3 4 5] is the permutation minimizing the feedback arc set. Let σb = [1 2 3 4 5 6]. Note that \|Eσa\|= 2, θ(σa,T) = \|{1,2}\|= 2. However \|Eσb\|= 3, yet θ(σb,T) = \|{6}\|= 1. Thus, σb gives a tighter upper bound on the dimension needed to represent T. 8 A PPLICATIONS OF THE RESULTS 8.1 C ONNECTIONS TO SIGN RANK The sign rank of a matrix G ∈{+1,−1}m×n is deﬁned as the smallest integer dsuch that there exists a matrix M ∈Rm×n of rank dthat satisﬁes sign(M) = G1. Here sign(z) = 1 if z >0 and −1 otherwise. A breakthrough result on the lower bound on the sign rank was given by Forster & Simon (2006). However, good upper bounds have been harder to obtain. We show below how Theorem 11 also translates to an upper bound on the sign rank of any sign matrix G. Theorem 12. Let G ∈{+1,−1}m×n be an arbitrary sign matrix. Let T be any tournament on m+ nnodes. Let S = {1,...,m }and let edge orientations of the cut(S, ¯S) in T be determined by G Then sign-rank(G) ≤2(µ(T) + 1) We argue that the the above result can give signiﬁcantly tigher bounds than the bounds in Alon et al. (2016). The simplest example one can consider is a locally transitive tournament onnnodes. Viewing this as a sign pattern matrix, it is easy to argue that the VC dimension of such a matrix is atmost 2. Theorem 5 in Alon et al. (2016) shows that the sign rank is upper bounded by O(√n)) when VC dimension is 2 (For a general hypothesis class of VC dimension d, the upper bound is O(n1−1 d). On the other hand, by deﬁnition µ(T) for any locally transitive tournament is exactly 0 (and so the upper bound of 2µ(T) + 1 is exactly 2 and is tight) for any locally transitive tournament which is independent of the number of nodes n. The important reason for this signiﬁcantly improved bound using µ(T) is that we consider only skew symmetric sign pattern matrices while Alon et al. (2016) look at the worst case (w.r.t representation dimension/sign rank) sign pattern matrices for a ﬁxed VC dimension. Our bound reduces the study of sign rank to a more graph theoretic study of the feedback arc set problem. It is not clear if the upper bound of 2(µ(T) + 1)can be improved and we leave this to future work. 8.2 C ONNECTIONS TO LEARNING FROM PAIRWISE COMPARISONS Consider a learning to rank problem from pairwise comparisons. Here, a set of nitems need to be ranked from a subset of pairwise comparisons among them. Every pair that is chosen for comparison is compared a ﬁxed number of times. Every time items iand jare compared, iis preferred over j 1Sign-rank can be deﬁned for any general matrix G ∈ Rm×n. We restrict to the sign matrices to make the connections to the tournament matrices explicit 8 Published as a conference paper at ICLR 2022 with probability Pij. A common and popular model to capture these probabilities is the Bradley- Terry-Luce (BTL) model Bradley & Terry (1952) wherePij = si/(si + sj) for some score vector s ∈Rn. Note that the model is completely speciﬁed by the score vector s which in turn completely determines the probability preference matrix P. The learning problem is to learn these unknown score vector s ∈Rn from noisy pairwise comparisons. Once the score vector is learnt, a ranking can be obtained by sorting these scores and the pairwise probabilities of unseen pairs of items can be predicted. The above BTL model is an example of a rank-2 model in the sense that the probability preference matrix P results in a rank 2 matrix under the log-odds skew symmetric transformation. Indeed, if we deﬁne Mij = log( Pij Pji ). then equivalently Mij = log(si) −log(sj). It is easy to show that M is a rank 2 skew symmetric matrix. However, the major disadvantage of the BTL model is that it can capture only transitive preferences i.e, Pij > 0.5 and Pjk > 0.5 = ⇒ Pki > 0.5. In real world situations, intransitivity is very common. To achieve intransitivity, the simplest way would be to start with a general rank rskew symmetric matrix M and consider the probability matrix that determines the preference probabilities as Pij = 1/(1 + exp(−Mij)). Here, the learning problem would be to estimate the two score vectors or equivalently the two dimensional representation for each item. Previous studies (Rajkumar & Agarwal (2016)) show that this can be learnt using Matrix completion based approaches or maximum likelihood based approaches (Chen & Joachims (2016)). However, what was not known earlier is the structure of tournaments that can be captured using such low rank restrictions which is important to decide on the parameter r. Our work helps modellers make informed decision to choose rbased on explicitly identifying the types of tournaments that can be modelled and subsequently learnt. We discuss next a simple application where modeling a ranking from pairwise comparisons problem can beneﬁt from the insights gained using our results. Consider a situation of modelling sports tournaments such as Tennis. Here, one can choose to model the players (nodes) using 2-dimensions where the dimensions corresponds to their offense (forehand) and defense (backhand) strengths respectively. When two players compete, the advantage of the offense strength of player1 w.r.t the defense of player 2 and vice versa determine the outcome of the match. This is precisely captured by a rank 2 model where the learning problem would be to infer these latent offense/defense strength of each player from outcomes of pairwise competitions. Theorem 6 says that such a model would immediately lead to locally transitive tournaments among the players. This structural characterisation now gives insights to the modeller if 2 dimensions are enough to model the players or not. 9 R EAL WORLD TOURNAMENT EXPERIMENTS We conducted simple experiments on real world data sets. Speciﬁcally, we considered 114 real world tournaments that arise in several applications including election candidate preferences, Sushi preferences, cars preferences, etc (source: www.preflib.org). The number of nodes in these tournaments varied from 5 to 23. Out of the 114 tournaments considered, 76(66.67%) were in fact locally transitive. For these tournaments, the upper bounds and lower bounds given by our theorems matched and was equal to 2. Interestingly, even for the non-locally transitive tournaments, the lower bound still turned out to be 2. We computed the upper bound for tournaments of size at most 9 (we did not do it for larger tournaments as this involves a brute force search) and found the value to be either 4 or 6. This shows that the upper bounds are usually non-trivial and efﬁciently approximating it is an interesting direction for future work. 10 C ONCLUSION In this work, we develop a theory of tournament representations. We show how ﬁxing the representa- tion dimension enforces, via forbidden conﬁgurations, restrictions on the type of tournaments that can be represented. We study and characterize rank 2 tournaments and show forbidden sub-tournaments for the rank d tournament class. We develop upper and lower bounds for minimum dimension needed to represent a tournament. Future work includes attempting to look deeper into some of the conjectures presented and possible strengthening of some of the bounds presented. 9 Published as a conference paper at ICLR 2022 REFERENCES Noga Alon, Shay Moran, and Amir Yehudayoff. Sign rank versus vc dimension. In Conference on Learning Theory, pp. 47–80. PMLR, 2016. Shalabh Bhatnagar, Michael C Fu, Steven I Marcus, and I-Jeng Wang. Two-timescale simulta- neous perturbation stochastic approximation using deterministic perturbation sequences. ACM Transactions on Modeling and Computer Simulation (TOMACS), 13(2):180–209, 2003. Amanda Bower and Laura Balzano. Preference modeling with context-dependent salient features. In International Conference on Machine Learning, pp. 1067–1077. PMLR, 2020. Ralph Allan Bradley and Milton E Terry. Rank analysis of incomplete block designs: I. the method of paired comparisons. Biometrika, 39(3/4):324–345, 1952. David Causeur and François Husson. A 2-dimensional extension of the Bradley–Terry model for paired comparisons. Journal of statistical planning and inference, 135(2):245–259, 2005. Pierre Charbit, Stéphan Thomassé, and Anders Yeo. The minimum feedback arc set problem is NP-hard for tournaments. Combinatorics, Probability and Computing, 16:01–04, 2007. Shuo Chen and Thorsten Joachims. Modeling intransitivity in matchup and comparison data. In Proceedings of the ninth acm international conference on web search and data mining, pp. 227–236, 2016. Nir Cohen, Marlio Paredes, and Sofía Pinzón. Locally transitive tournaments and the classiﬁcation of {(1,2)}-symplectic metrics on maximal ﬂag manifolds. Illinois Journal of Mathematics, 48(4): 1405–1415, 2004. David C Fisher and Jennifer Ryan. Tournament games and positive tournaments. Journal of Graph Theory, 19(2):217–236, 1995. Jürgen Forster. A linear lower bound on the unbounded error probabilistic communication complexity. 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Psychological review, 76(1):31, 1969. 10 Published as a conference paper at ICLR 2022 A A PPENDIX Proof of Theorem 1 Proof. Assume there is a T′ ∈F(T) which is not forbidden for rank dtournament class. Thus, ∃H′= {h′ 1,..., h′ n}∈ Rd such that T′= T[H′]. By deﬁnition, there must also exist a S ⊆[n] such that T′= φS(T). Consider the representation H obtained from H′where hi = −h′ i for all i ∈S and hi = h′ i otherwise. It is easy to verify that T = T[H] which is a contradiction to the assumption that T is a forbidden conﬁguration for rank dtournaments. Proof of Proposition 1 Proof. Consider any tournament T. Let T′= φ{i∪T− i }(T) for an arbitrary node i. By deﬁnition T′∈F(T). Also, T′is a R-cone, coned by i. Proof of Lemma 2 Proof. Consider any representation H that induces a tournament R. By assumption of the theorem, H positively spans Rd. Thus, by Farkas’ lemma, there cannot exist av ∈Rd such that vThi >0 ∀i. Note that if R-cone is not a forbidden conﬁguration for rank dtournaments, then there must exist a h ∈Rd such that hTArothi > 0 ∀i. As Arot is invertible, one can set v = ( Arot)Th with the property that vThi > 0 ∀i. But this contradicts the conclusion drawn earlier from Farka’s lemma. Proof of Proposition 2 Proof. This is easily veriﬁed by checking all tournaments inF(T) where T is the coned 3-cycle. Proof of Lemma 3 Proof. Wlog, assume that 1 ≥T 2 ≥T 3 ≥T 1. Then, it must be the case that the counterclockwise angle between the representation of the corresponding items must be ≤180 degrees. However, if the representations {h1,h2,h3}did not positively span R2, then by Farka’s Lemma, there must be some supporting hyperplane for the representations. However, this would imply at least one of the node pairs {(1,2),(2,3),(3,1)}must necessarily make an angle ≥180 degrees. But this contradicts the assumption that the nodes form a 3 cycle. Proof of Theorem 5 Proof. Let T be a transitive tournament on nnodes and let T′∈F(T). Then, there exists some S ⊆[n] such that T′ = φS(T). Consider any node i ∈[n]. Deﬁne following 4 subset of nodes associated with i: S1 = T+ i ∩S,S2 = T+ i \S1,S3 = T− i ∩S,S4 = T− i \S. Note that each of T(Sk) for k = 1 to 4 is a transitive sub-tournament and the relationship across Si,Sj for any two sets is either one completely beats the other or completely loses to the other. Note that in φS(T) the orientation of the edges across these sets is either ﬂipped as a whole or not ﬂipped at all. Thus, exactly two of these sets will be part of T′+ i and two part of T− i (the exact sets among these sets will depend on whether i∈Sor not), thus preserving the local transitivity property. Thus T′is locally transitive. To prove the opposite direction, letT be a locally transitive tournament. Let i∈[n] be an arbitrary node. We argue that T′:= φT+ i (T) is a transitive tournament. We will show this by arguing that there does not exist a 3 cycle in T′. Consider any 3-cycle a >T b >T c >T a. As T is locally transitive, not all {a,b,c }can be in T+ i . Also not all {a,b,c }can avoid T+ i as [n] \T+ i is transitive. Thus at least one and at most two of{a,b,c }belongs to T+ i (T). This means that the 3-cycle becomes a transitive tournament in T′:= φT+ i (T). Thus every cycle in T becomes transitive in T′. Now consider any 3 nodes which forms a transitive tournament a >T b >T cand involves at least one 11 Published as a conference paper at ICLR 2022 node and at most two nodes in T+ i . Then there are only two cases to consider: (1) a ∈[n] \T+ i and {b,c}∈ T+ i or (2) {a,b}∈ [n] \T+ i and c∈T+ i . In both these cases, it is easy to verify that the the corresponding tournament in T′is either the transitive tournament b>T′>c> T′ aor the transitive tournament c>T′ a>T′ b. As these are the only possibilities, the result follows by noting that T′= φT+ i (T) =⇒ T = φT+ i (T′) and so T ∈F(T′). Proof of Theorem 6 Proof. Assume T is locally transitive. Then by Theorem 5, it must be in the ﬂip class of some transitive tournament T′i.e. T = φS(T′) for some S ⊆[n]. It is easy to represent a transitive tournament using a rank 2 skew symmetric matrix. Indeed pick any vector u ∈ Rn which is sorted according to the topological ordering of T′. Let v ∈Rn be the all ones vector. Then M = uvT −vuT represents T′. Let M = (H′)TArotH for some H′∈R2×n.Then the columns of H′represent T′. Now consider the representation H′obtained from H′where the columns indexed by Sare multiplied by −1. This does not change the rank of H′and it can be veriﬁed that T = T[H]. To prove the other direction, consider any rank 2 skew symmetric matrix M ∈Rn×n. Then there must exist u,v ∈Rn such that M = uvT −vuT. Consider any node i∈[n]. Consider three nodes a,b,c such that uivj >ujvi for j ∈{a,b,c }. Furthermore let uavb >vaub and ubvc >vbuc. Then by carefully going over all {±}sign possibilities for {ui,ua,ub,uc,vi,va,vb,vc}, one can conclude that it must be the case that uavc >vauc. This just shows that T+ i is transitive. Analogously one can show that T− i is also transitive. As iwas arbitrary, the result follows. Proof of Theorem 7 Proof. Recall that the classic quick sort algorithm picks a pivot node (say 1) and places all nodes that beat the pivot to the right in the ranking and those thatlose to the left and then recurses on the left and right subsets. As T is locally transitive, choosing any pivot iwould correspond to ﬁxing the pivots position and simply returning the ranking [σ(N− i ),i,σ (N+ i )] where σ(N+ i ), σ(N− i ) correspond to the topological ordering of the transitive tournamentsT(N+ i ) and T(N− i ) respectively. We ﬁrst argue that changing the pivot only cyclically shifts the ﬁnal ranking. To see why this is true, consider two pivots iand jand their corresponding rankings σi and σj. Without loss of generality, assume that the ranking σi = [1,...,n ] and i>T j. We argue that there exists an integer ksuch that N+ j = {j+ 1 mod n,(j+ 2) mod n,..., (j+ k) mod n}(where by convention n mod n = n). As N+ i is transitive, and jis part of it, it must be the case that all the nodes {j+ 1,...,j + n}∈ N+ j . Then to prove the claim, it remains to be shown that the set N− i ∩N+ j is either empty or must be the nodes {1,...,ℓ }for some ℓ<i . If empty, we are done. If not, assume for the sake of contradiction that there exists three succesive integers ℓa,ℓb,ℓc < isuch that ℓa,ℓc <T j but ℓb >T j. It is easy to verify that this cannot happen as it would lead to T({ℓa,ℓb,ℓc,j}) being a forbidden conﬁguration. Notice that as every locally transitive tournament is in the ﬂip class of a transitive tournament i.e., T = φS(T) for some S, one can divide the set of all nodes into 2k+ 1 groups as follows: Let [1,...,n ] be the ordering corresponding to the transitive tournament wlog. Starting from 1, add as many nodes to a group such that all the elements belong to either S or ¯S. Once the condition is violated, create a new group and continue the same process. It is not hard to verify that 2k+ 1 groups will be formed in this process for some k ≥0. Moreover, each group would separate two other groups by construction. We can ﬁrst show that the items of a single group must appear in consecutive positions in one of the optimal rankings. This is proven as follows. Consider there exists an optimal ranking with items which belong to the same group not occurring consecutively. Consider two items belonging to the same group, which have items from other groups present in between them in the ranking. Consider these items to be a1,a2, with a1 present above in the rankings. Consider the number of upsets that the two items are involved in to be u1 and u2. If u1 ≤u2, a2 can be placed right after a1 in the ranking, creating a better or equivalent ranking in terms of upsets. Similarly if u1 ≥u2, a1 can be placed directly above a2 in the rankings to create 12 Published as a conference paper at ICLR 2022 an equivalent or better ranking. Therefore there exists an optimal ranking which has all items in the same group consecutively. This theorem is then reduced to ﬁnding a ranking of groups, which is proven using induction on k. Base Case Consider the base case with k= 1. Let there be 3 groups, C, A1, B1. We can say that the optimal ranking cannot be any of the following CB1A1 A1CB1 B1A1C since all three rankings can be made better by swapping the second and third ranked groups. Therefore the 3 possible optimal rankings are CA1B1 A1B1C B1CA1 which are cyclic shifts of each other. Inductive Step One property of rankings which is useful for the inductive step proof is as follows. Let there be2k+ 1 groups G= {g1,g2 ...g 2k+1}. Label the optimal ranking with the condition that gi be placed ﬁrst in the ranking as Ri. The ranking Ri with gi removed must be the optimal ranking for G\{gi}. This can be shown using contradiction i.e, if there was a better ranking for G\gi, that ranking with gi appended to the front would be better than Ri. We now assume the theorem is true for size 2k−1 instances and aim to prove for the same for size 2k+ 1 instances. Consider g1 as the ﬁrst group in the ranking. This creates a certain number of upsets, for the purposes of ranking the remaining groups, 2 of the remaining groups can be merged into a single group. This follows from the observation earlier that each group also ’separates’ two groups. This can be considered an instance of the size 2k−1 problem. Therefore the set of optimal rankings with g1 as the ﬁrst group in the rankings is made up of g1 as the ﬁrst group and a cyclic sweep of the remaining items to ﬁll the remaining positions. Therefore the optimal permutation must be among the sets created by considering each of the 2k+ 1 groups as the ﬁrst group in the rankings. Let Ri,j represent the ranking which has group gi as the ﬁrst group and the remaining groups present as a cyclic sweep from gj. Consider the case of R1,k. Let xi represent the number of items in group gi. If R1,k is a better ranking than Rk,k+1, it implies that n+1∑ i=k xi > 2n+1∑ i=n+2 xi (1) by considering the shift of g1 in the two rankings. The difference in the number of upsets between R1,2 and R1,k is given by x2(−xk−xk+1 ... −xn+2 + xn+3 ...x 2n+1) +x3(−xk−xk+1 ... −xn+3 + xn+4 ...x 2n+1) ... Using Equation 1, it can be seen that each of the above terms are negative for any j ≤n+ 1, making R1,2 the better ranking. Any j >n+ 1 cannot be considered as an optimal ranking since the ﬁrst group as per the ranking must precede the second(otherwise switching them would decrease the upsets). Since either R1,2 or Rk,k+1(both counterclockwise orderings) is better than R1,k whenever k≤n+ 1(R1,k cannot be the optimal ranking when k >n+ 1), and since this can be generalised for any Ri,j, it is shown that one of the counterclockwise orderings of the items is the optimal ranking. We note that the above proof also appeared in ?. However, the insights via ﬂip classes were not present there. 13 Published as a conference paper at ICLR 2022 A.0.1 F INDING FORBIDDEN CONFIGURATIONS Given a tournament and a representation dimension, there is no known method to check whether the tournament can be represented by vectors in the given dimension. The forbidden conﬁgurations presented above were found by carefully creating an exhaustive set of cases, and showing each one causes a contradiction. The techniques used are presented below. Proof of Theorem 8 Proof. Consider the representation of tournaments as given in Subsection 3. Since adding minor noise to each hi will not change tournament, we can deal only with cases in which any size 4 subset of h1,h2,...h12 consists of linearly independent vectors. By using this property, and without loss of generality, c1h1 + c2h2 + c3h3 + c4h4 = h5 (2) We can construct cases based on the signs of the coefﬁcients(ci) in the above equation. There are 24 = 16 sign patterns/cases for any equation. These cases are ﬁltered by multiplying the entire expression with expressions of the form Arothi. c1h1Arothi + c2h2Arothi + c3h3Arothi + c4h4Arothi = h5Arothi (3) In the above equation, the signs of all expressions of the form hiArothj is known from the tournament conﬁguration. Therefore simply comparing the sign of the LHS and RHS for all possible values of irules out many cases. We now use multiple equations together in a bid to further ﬁlter the remaining cases. Consider the following equations without loss of generality. c1h1 + c2h2 + c3h3 + c4h4 = h5 (4) b1h2 + b2h3 + b3h4 + b4h5 = h6 (5) a1h1 + a2h2 + a3h3 + a4h4 = h6 (6) h5 can be eliminated from the ﬁrst two equations, leaving two expressions ofh6 in terms of h1,...h 4. The two sets of coefﬁcients of h1,...h 4 can be equated, and sign based arguments can eliminate a few more cases. Also, a set of 3 tuples can be constructed, with each item representing possible sign patterns for the 3 equations. Note that this set of 3 tuples does not contain entries which cannot apply simultaneously on the 3 equations. The above elimination step can be considered as a ﬁltration procedure given a size 6 tuple (h1,h2,h3,h4,h5,h6) by using 3 equations. We can perform a similar ﬁltration step given any size 6 tuple as well. Let the equations corresponding to (h1,h2,h3,h4,h5,h6) be E1,1,E1,2,E1,3. Similarly, consider the tuples(h1,h2,h3,h4,h5,h7), (h2,h3,h4,h5,h6,h7), (h1,h2,h3,h4,h6,h7) and their corresponding equations represented by Ei,j where iis the index of the tuple it corresponds to and jis the index of the equation. The above ﬁltration process can be performed on all the tuples, following which an additional ﬁltering step can be performed by using • E1,1 ≡E2,1 • E1,2 ≡E3,1 • E1,3 ≡E4,1 • E2,2 ≡E3,3 • E2,3 ≡E4,2 where the equivalence relation represents that the 2 equations are identical. Since identical equations must have identical sign patterns, the sign patterns not present in both sets of possible sign patterns can be ﬁltered out. For the 11-DRT cone, this leaves us with null set, proving that it is a forbidden conﬁguration. 14 Published as a conference paper at ICLR 2022 Proof of Theorem 9 Proof. The result follows the arguments in Forster & Simon (2006). We note that the arguments in Forster & Simon (2006) work only for non-zero sign matrices. By overloading T to denote the signed adjacency matrix of the corresponding tournament, we can consider the non-zero sign matrix G = T + diag(b) where b ∈{1,−1}n. By Gershgorin’s circle theorem and exploting the fact that any two rows of a doubly regular tournament are orthogonal, one can show that ρi(G)2 ≤ρi(T)2 + 2n−2 where ρi denotes the i-th largest singular value of the corresponding matrix. It then follows from Forster & Simon (2006) that if G has an representation in dim dimensions, then it must be the case that dim dim∑ i=1 (ρi(G))2 ≥n2. Noting that for a doubly regular cone T on nnodes, ρi(T).2 = n−1 ∀i, we get dim2 ·3(n−1) ≥n2 =>dim ≥ √n 3 (7) Thus to get a matrixM such that sign(M) = G, one needs at least√n 3 dimensions i.e., rank(M) ≥√n 3 . Now we show that this also is a lower bound for representing T. To see this, assume for the sake of contradiction that T has a representation in d dimension where d <√n 3 . Then, there exists H ∈Rd×n such that T = T[H]. The diagonal entries of T[H] are zero. Now introduce a small enough perturbation E to H and consider the matrix (H + E)TArotH. The entries of the perturbation matrix E ∈Rd×n can be chosen to be small enough such that the sign of the off-diagonal entries of HTArotH is same as that of (H + E)TArotH. However, the diagonal entries of (H + E)TArotH can get an arbitrary sign pattern, say b. Let G be the sign pattern matrix corresponding to (H + E)TArotH. By deﬁnition, G has a representation in dimension d< √n 3 . But this contradicts inequality (7). Thus, it must be the case that T has a representation in dimension at least √n 3 . Finally, setting n= 12d2 to the number of nodes in the doubly regular cone as given in the Theorem, we get that at least 2ddimensions are needed to embed such a tournament. The result follows. Proof of Theorem 10 Proof. The proof is the same arguments as the ﬁrst part of the proof of Theorem 9. Proof of Theorem 11 Proof. Given a tournament T, we ﬁrst show that we can start with an arbitrary transitive tournament and add enough rank 2 corrections to obtain a representation for T. Every addition of a rank 2 matrix will increase the representation dimension by at most 2. The result will follow then by noting that for the choice of the transitive tournament which minimizes the number of corrections needed, one needs at most 2(µ(T) + 1)representation dimension. Let T′ be an arbitrary transitive tournament which has a 2 dimensional representation and let M ∈Rn×n be the associated skew symmetric matrix that represents T′. W.l.o.g, assume that Mij >0 if and only if i<j . Deﬁne Ek(T) = {i: i<k, i< T k}. By deﬁnition, the feedback arc set Eσ(T) = ∪n k=1(k,Ek(T)) where σ = [1,...,n ] is the topological order corresponding to the transitive tournament T′. We start with k= nand correct the feedback arc errors arising from Ek iteratively. Deﬁne ∆n = √ max i<n (Min + ϵ) for some small enough ϵ> 0. Let u ∈Rn be such that ui = ∆ ∀i∈En and ui = 0 otherwise. Let v ∈Rn be such that vn = −∆,vi = 0 ∀i̸= n. It is easy to verify that M+uvT −vuT represents a tournament that has the feedback arc set∪n−1 k=1 (k,Ek(T)) i.e., the errors in En have been corrected. The cost of correcting the error is adding a rank 2 skew 15 Published as a conference paper at ICLR 2022 symmetric matrix which increases the representation dimension by at most 2. One can repeat the same procedure for n−1,n −2,... until all errors are corrected. The upper bound in the theorem follows noting that the above argument works for any transitive tournament T′and so we can start with the one which has the least number of nodes involved in the feedback arc set to minimize the number of extra dimensions needed to represent T. Proof of Theorem 12 Proof. From Theorem 11, T can be represented using at most 2(µ(T) + 1)dimensions. By construc- tion any representation of T must also represent G. The result follows. A.0.2 A DDITIONAL FILTRATION FOR VERIFYING 10 N ODE FORBIDDEN CONFIGURATION T =   0 1 1 1 −1 −1 −1 −1 −1 1 −1 0 1 −1 1 1 −1 −1 1 −1 −1 −1 0 1 1 −1 1 −1 −1 −1 −1 1 −1 0 −1 1 1 −1 −1 −1 1 −1 −1 1 0 1 −1 −1 1 −1 1 −1 1 −1 −1 0 1 −1 1 −1 1 1 −1 −1 1 −1 0 −1 1 −1 1 1 1 1 1 1 1 0 1 −1 1 −1 1 1 −1 −1 −1 −1 0 −1 −1 1 1 1 1 1 1 1 1 0   Due to the lesser number of constraints present when dealing with 10 nodes instead of 12, an additional ﬁltration step is required. The entire procedure described above can be considered as an elimination procedure given a tuple T1 = (h1,...h 7). Also consider that in the ﬁnal ﬁltration above, a set of size 4 tuples is created. These size 4 tuples will represent the possible sign patterns for E1,1,E1,2,E1,3,E2,2. If a similar procedure is carried out on T2 = (h2,...h 8), another set of 4 tuples will be generated. The ﬁltration in this step is based on the fact that the equations E1,3,E2,2 corresponding to T1 are the same as E1,1,E1,2 for T2. The sign patterns which are not present in both sets of tuples are ﬁltered out. This type of relationship can be obtained between T2 and T3 = (h3,...h 8,h0) as well, and in general between any Ti and Tj such that Tj can be obtained from Ti by removing the ﬁrst entry and adding an entry to the end. Note that the tuples must have unique entries. Therefore a ’circular elimination’ procedure can be followed, where the Ti tuples considered are size 7 subsets of the circular permutations of h1,...h 8. By following this procedure, all the sign patterns remaining can be eliminated. Using size 7 circular permutations of h1,...h 10 in the last step does not eliminate all the cases, we believe this is due to nodes 1 to 8 forming a 7-DRT cone, which leads to many eliminations. The code for the ﬁltration can be found here. 16	Arun Rajkumar, Vishnu Veerathu, Abdul Bakey Mir	Accept (Poster)	2,022	{"id": "zzk231Ms1Ih", "original": "_vWu7gLdTrwb", "cdate": 1632875730339, "pdate": 1643407560000, "odate": 1633539600000, "mdate": null, "tcdate": 1632875730339, "tmdate": 1676330471169, "ddate": null, "number": 4095, "content": {"title": "A Theory of Tournament Representations", "authorids": ["~Arun_Rajkumar4", "~Vishnu_Veerathu1", "cs20d400@smail.iitm.ac.in"], "authors": ["Arun Rajkumar", "Vishnu Veerathu", "Abdul Bakey Mir"], "keywords": ["tournament", "skew-symmetric", "pairwise ranking"], "abstract": "Real-world tournaments are almost always intransitive. Recent works have noted that parametric models which assume $d$ dimensional node representations can effectively model intransitive tournaments. However, nothing is known about the structure of the class of tournaments that arise out of any fixed $d$ dimensional representations. In this work, we develop a novel theory for understanding parametric tournament representations. Our first contribution is to structurally characterize the class of tournaments that arise out of $d$ dimensional representations. We do this by showing that these tournament classes have forbidden configurations that must necessarily be a union of flip classes, a novel way to partition the set of all tournaments. We further characterize rank $2$ tournaments completely by showing that the associated forbidden flip class contains just $2$ tournaments. Specifically, we show that the rank $2$ tournaments are equivalent to locally transitive tournaments. This insight allows us to show that the minimum feedback arc set problem on this tournament class can be solved using the standard Quicksort procedure. We also exhibit specific forbidden configurations for rank $4$ tournaments. For a general rank $d$ tournament class, we show that the flip class associated with a coned-doubly regular tournament of size $\\mathcal{O}(\\sqrt{d})$ must be a forbidden configuration. To answer a dual question, using a celebrated result of Froster, we show a lower bound of $\\Theta(\\sqrt{n})$ on the minimum dimension needed to represent all tournaments on $n$ nodes. For any given tournament, we show a novel upper bound on the smallest representation dimension that depends on the least size of the number of unique nodes in any feedback arc set of the flip class associated with a tournament. We show how our results also shed light on the upper bound of sign-rank of matrices. ", "one-sentence_summary": "We develop a theory to understand tournament representations i.e. structurally characterise when a tournament graph can be represented in lower dimensions using a skew symmetric matrix. ", "code_of_ethics": "", "submission_guidelines": "", "resubmission": "", "student_author": "", "serve_as_reviewer": "", "paperhash": "rajkumar\|a_theory_of_tournament_representations", "pdf": "/pdf/a7853d8c301f8a37bc858f4c428d73862dabff26.pdf", "_bibtex": "@inproceedings{\nrajkumar2022a,\ntitle={A Theory of Tournament Representations},\nauthor={Arun Rajkumar and Vishnu Veerathu and Abdul Bakey Mir},\nbooktitle={International Conference on Learning Representations},\nyear={2022},\nurl={https://openreview.net/forum?id=zzk231Ms1Ih}\n}", "venue": "ICLR 2022 Poster", "venueid": "ICLR.cc/2022/Conference"}, "forum": "zzk231Ms1Ih", "referent": null, "invitation": "ICLR.cc/2022/Conference/-/Blind_Submission", "replyto": null, "readers": ["everyone"], "nonreaders": [], "signatures": ["ICLR.cc/2022/Conference"], "writers": ["ICLR.cc/2022/Conference"]}	[Main Review]: Tournaments are important tools in sports modeling, social preference, etc. A tournament on n nodes is the complete directed graph where every pair of nodes has a directed edge pointing from one vertex to the other. One can represent a tournament as the sign matrix of a rank d matrix. In this work, the authors develop a novel theory for understanding tournament representation. Firstly, the authors characterize the structure of rank d tournaments using a list of forbidden configurations. These forbidden configurations are formed by the union of so-called flip classes. For rank 2 tournaments, the authors use this framework to describe precisely the forbidden configurations and show that they are actually equivalent to locally transitive tournaments, a class of tournaments closely related to transitive tournaments. The authors also show one forbidden flip class for rank 4 tournaments and also a weaker forbidden flip class for the general rank d case. Secondly, the authors show both lower and upper bound on the minimum dimension required for tournament representation on n nodes. The lower bound of order sqrt(n) is proved by a variation of the celebrated work of Forster and Simon for sign matrices. The upper bound is given in terms of a parameter named the Flip Feedback Node set of a Tournament introduced by the authors. Such an upper bound also gives new bounds for sign ranks of matrices. The results of this paper are overall nice, but I find some of results in the first part slightly weak. Especially for rank-4 and rand-d tournaments the forbidden flip classes are still not totally clear. Also, it is unclear to me if the minimum feedback arc set problem can be solved efficiently for constant rank representable tournaments. Is this problem efficiently solvable when the representation matrix is given? In fact, more discussions about applications of low-rank representations or forbidden flip classes would be appreciated. As a potential direction for improvement or future direction, one can probably also try studying the representation dimension of random tournaments. Detailed comments: Page 2: “We exhibit a lower bound of O(\sqrt(n))” Technically you should use \Omega() for lower bound instead of big-O Page 4, Definition 3: I think in the first item you should have “for all j \neq i^*” instead of i	ICLR.cc/2022/Conference/Paper4095/Reviewer_AhwX	null	3	{"id": "mvJh08SL8ke", "original": null, "cdate": 1635890975327, "pdate": null, "odate": null, "mdate": null, "tcdate": 1635890975327, "tmdate": 1635890975327, "ddate": null, "number": 2, "content": {"summary_of_the_paper": "This paper provides fundamental theories of tournament representations. The authors study two main questions. First they characterize the class of tournaments that can be represented in d dimensions. Second they give lower and upper bounds on the minimum dimension needed to represent a tournament on n nodes.\n", "main_review": "Tournaments are important tools in sports modeling, social preference, etc. A tournament on n nodes is the complete directed graph where every pair of nodes has a directed edge pointing from one vertex to the other. One can represent a tournament as the sign matrix of a rank d matrix. In this work, the authors develop a novel theory for understanding tournament representation. \n\nFirstly, the authors characterize the structure of rank d tournaments using a list of forbidden configurations. These forbidden configurations are formed by the union of so-called flip classes. For rank 2 tournaments, the authors use this framework to describe precisely the forbidden configurations and show that they are actually equivalent to locally transitive tournaments, a class of tournaments closely related to transitive tournaments. The authors also show one forbidden flip class for rank 4 tournaments and also a weaker forbidden flip class for the general rank d case.\n\nSecondly, the authors show both lower and upper bound on the minimum dimension required for tournament representation on n nodes. The lower bound of order sqrt(n) is proved by a variation of the celebrated work of Forster and Simon for sign matrices. The upper bound is given in terms of a parameter named the Flip Feedback Node set of a Tournament introduced by the authors. Such an upper bound also gives new bounds for sign ranks of matrices.\n\nThe results of this paper are overall nice, but I find some of results in the first part slightly weak. Especially for rank-4 and rand-d tournaments the forbidden flip classes are still not totally clear. Also, it is unclear to me if the minimum feedback arc set problem can be solved efficiently for constant rank representable tournaments. Is this problem efficiently solvable when the representation matrix is given? In fact, more discussions about applications of low-rank representations or forbidden flip classes would be appreciated. As a potential direction for improvement or future direction, one can probably also try studying the representation dimension of random tournaments. \n\nDetailed comments:\nPage 2: \u201cWe exhibit a lower bound of O(\\sqrt(n))\u201d\nTechnically you should use \\Omega() for lower bound instead of big-O\nPage 4, Definition 3: I think in the first item you should have \u201cfor all j \\neq i^*\u201d instead of i\n", "summary_of_the_review": "Overall this is a nice paper though I think there is still room for improvements for some parts. Some results are slightly weak and I don\u2019t really see any application for most results. Perhaps the authors should discuss more about the application parts.\n", "correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "Not applicable", "flag_for_ethics_review": ["NO."], "recommendation": "5: marginally below the acceptance threshold", "confidence": "3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked."}, "forum": "zzk231Ms1Ih", "referent": null, "invitation": "ICLR.cc/2022/Conference/Paper4095/-/Official_Review", "replyto": "zzk231Ms1Ih", "readers": ["everyone"], "nonreaders": [], "signatures": ["ICLR.cc/2022/Conference/Paper4095/Reviewer_AhwX"], "writers": ["ICLR.cc/2022/Conference", "ICLR.cc/2022/Conference/Paper4095/Reviewer_AhwX"]}	{ "criticism": 3, "example": 1, "importance_and_relevance": 2, "materials_and_methods": 5, "praise": 2, "presentation_and_reporting": 2, "results_and_discussion": 3, "suggestion_and_solution": 4, "total": 19 }	1.157895	0.589727	0.568167	1.175681	0.207894	0.017787	0.157895	0.052632	0.105263	0.263158	0.105263	0.105263	0.157895	0.210526	{ "criticism": 0.15789473684210525, "example": 0.05263157894736842, "importance_and_relevance": 0.10526315789473684, "materials_and_methods": 0.2631578947368421, "praise": 0.10526315789473684, "presentation_and_reporting": 0.10526315789473684, "results_and_discussion": 0.15789473684210525, "suggestion_and_solution": 0.21052631578947367 }	1.157895	iclr2022	openreview	0	0			0	null
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zzk231Ms1Ih	A Theory of Tournament Representations	"Real-world tournaments are almost always intransitive. Recent works have noted that parametric mode(...TRUNCATED)	"Published as a conference paper at ICLR 2022\nA THEORY OF TOURNAMENT REPRESENTATIONS\nArun Rajkumar(...TRUNCATED)	Arun Rajkumar, Vishnu Veerathu, Abdul Bakey Mir	Accept (Poster)	2,022	"{\"id\": \"zzk231Ms1Ih\", \"original\": \"_vWu7gLdTrwb\", \"cdate\": 1632875730339, \"pdate\": 1643(...TRUNCATED)	"[Main Review]:\nThe goal of connecting dimensional characterization and structural characterization(...TRUNCATED)	ICLR.cc/2022/Conference/Paper4095/Reviewer_UVYb	null	3	"{\"id\": \"C9NhLV0gKlP\", \"original\": null, \"cdate\": 1635972587363, \"pdate\": null, \"odate\":(...TRUNCATED)	{"criticism":4,"example":1,"importance_and_relevance":2,"materials_and_methods":2,"praise":4,"presen(...TRUNCATED)	2.1	1.957958	0.142042	2.111869	0.158905	0.011869	0.4	0.1	0.2	0.2	0.4	0.5	0.2	0.1	{"criticism":0.4,"example":0.1,"importance_and_relevance":0.2,"materials_and_methods":0.2,"praise":0(...TRUNCATED)	2.1	iclr2022	openreview	0	0			0	null
zyrhwrd9EYs	To Impute or Not To Impute? Missing Data in Treatment Effect Estimation	"Missing data is a systemic problem in practical scenarios that causes noise and bias when estimatin(...TRUNCATED)	"Under review as a conference paper at ICLR 2022\nTO IMPUTE OR NOT TO IMPUTE ? M ISSING DATA IN\nTRE(...TRUNCATED)	Jeroen Berrevoets, Fergus Imrie, Trent Kyono, James Jordon, Mihaela van der Schaar	Reject	2,022	"{\"id\": \"zyrhwrd9EYs\", \"original\": \"VHkY-hTHhg\", \"cdate\": 1632875675785, \"pdate\": 164340(...TRUNCATED)	"[Main Review]:\nStrengths of paper\n\n- Paper presents a very elaborate writeup with a detailed(...TRUNCATED)	ICLR.cc/2022/Conference/Paper3258/Reviewer_efJq	null	3	"{\"id\": \"vop8d7q0nC4\", \"original\": null, \"cdate\": 1636035619779, \"pdate\": null, \"odate\":(...TRUNCATED)	{"criticism":2,"example":0,"importance_and_relevance":3,"materials_and_methods":2,"praise":3,"presen(...TRUNCATED)	2	1.605439	0.394561	2.00806	0.183631	0.00806	0.25	0	0.375	0.25	0.375	0.125	0.25	0.375	{"criticism":0.25,"example":0.0,"importance_and_relevance":0.375,"materials_and_methods":0.25,"prais(...TRUNCATED)	2	iclr2022	openreview	0	0			0	null
zyrhwrd9EYs	To Impute or Not To Impute? Missing Data in Treatment Effect Estimation	"Missing data is a systemic problem in practical scenarios that causes noise and bias when estimatin(...TRUNCATED)	"Under review as a conference paper at ICLR 2022\nTO IMPUTE OR NOT TO IMPUTE ? M ISSING DATA IN\nTRE(...TRUNCATED)	Jeroen Berrevoets, Fergus Imrie, Trent Kyono, James Jordon, Mihaela van der Schaar	Reject	2,022	"{\"id\": \"zyrhwrd9EYs\", \"original\": \"VHkY-hTHhg\", \"cdate\": 1632875675785, \"pdate\": 164340(...TRUNCATED)	"[Main Review]:\nAuthors use graphic causal models to model the various missingness in causal effect(...TRUNCATED)	ICLR.cc/2022/Conference/Paper3258/Reviewer_8bft	null	4	"{\"id\": \"Sw15Q7x-eNY\", \"original\": null, \"cdate\": 1635330744624, \"pdate\": null, \"odate\":(...TRUNCATED)	{"criticism":8,"example":4,"importance_and_relevance":1,"materials_and_methods":3,"praise":1,"presen(...TRUNCATED)	1.75	0.976661	0.773339	1.76138	0.136165	0.01138	0.4	0.2	0.05	0.15	0.05	0.4	0.4	0.1	{"criticism":0.4,"example":0.2,"importance_and_relevance":0.05,"materials_and_methods":0.15,"praise"(...TRUNCATED)	1.75	iclr2022	openreview	0	0			0	null
zyrhwrd9EYs	To Impute or Not To Impute? Missing Data in Treatment Effect Estimation	"Missing data is a systemic problem in practical scenarios that causes noise and bias when estimatin(...TRUNCATED)	"Under review as a conference paper at ICLR 2022\nTO IMPUTE OR NOT TO IMPUTE ? M ISSING DATA IN\nTRE(...TRUNCATED)	Jeroen Berrevoets, Fergus Imrie, Trent Kyono, James Jordon, Mihaela van der Schaar	Reject	2,022	"{\"id\": \"zyrhwrd9EYs\", \"original\": \"VHkY-hTHhg\", \"cdate\": 1632875675785, \"pdate\": 164340(...TRUNCATED)	"[Main Review]:\nThe questions they are trying to solve is an important problem.\nFor example in the(...TRUNCATED)	ICLR.cc/2022/Conference/Paper3258/Reviewer_ZWmF	null	5	"{\"id\": \"-aEdXoBehI4\", \"original\": null, \"cdate\": 1634737253988, \"pdate\": null, \"odate\":(...TRUNCATED)	{"criticism":1,"example":1,"importance_and_relevance":1,"materials_and_methods":12,"praise":2,"prese(...TRUNCATED)	2	1.857958	0.142042	2.012438	0.180964	0.012438	0.0625	0.0625	0.0625	0.75	0.125	0.125	0.5	0.3125	{"criticism":0.0625,"example":0.0625,"importance_and_relevance":0.0625,"materials_and_methods":0.75,(...TRUNCATED)	2	iclr2022	openreview	0	0			0	null
zuqcmNVK4c2	Self-Joint Supervised Learning	"Supervised learning is a fundamental framework used to train machine learning systems. A supervised(...TRUNCATED)	"Published as a conference paper at ICLR 2022\nSELF -JOINT SUPERVISED LEARNING\nNavid Kardan†∗, (...TRUNCATED)	Navid Kardan, Mubarak Shah, Mitch Hill	Accept (Poster)	2,022	"{\"id\": \"zuqcmNVK4c2\", \"original\": \"LC1zzwgqdrA\", \"cdate\": 1632875666960, \"pdate\": 16434(...TRUNCATED)	"[Main Review]:\nStrength:\n\n1. The main idea of modelling the conditional joint label distribution(...TRUNCATED)	ICLR.cc/2022/Conference/Paper3122/Reviewer_Ky9n	null	3	"{\"id\": \"bgq04xhGww\", \"original\": null, \"cdate\": 1635898995369, \"pdate\": null, \"odate\": (...TRUNCATED)	{"criticism":4,"example":1,"importance_and_relevance":4,"materials_and_methods":13,"praise":5,"prese(...TRUNCATED)	1.285714	-2.265332	3.551047	1.30096	0.175397	0.015246	0.142857	0.035714	0.142857	0.464286	0.178571	0.107143	0.107143	0.107143	{"criticism":0.14285714285714285,"example":0.03571428571428571,"importance_and_relevance":0.14285714(...TRUNCATED)	1.285714	iclr2022	openreview	0	0			0	null
zuqcmNVK4c2	Self-Joint Supervised Learning	"Supervised learning is a fundamental framework used to train machine learning systems. A supervised(...TRUNCATED)	"Published as a conference paper at ICLR 2022\nSELF -JOINT SUPERVISED LEARNING\nNavid Kardan†∗, (...TRUNCATED)	Navid Kardan, Mubarak Shah, Mitch Hill	Accept (Poster)	2,022	"{\"id\": \"zuqcmNVK4c2\", \"original\": \"LC1zzwgqdrA\", \"cdate\": 1632875666960, \"pdate\": 16434(...TRUNCATED)	"[Main Review]:\nStrengths:\n1. Simple idea that works very well\n2. Paper is easy to understand\n3.(...TRUNCATED)	ICLR.cc/2022/Conference/Paper3122/Reviewer_PZCK	null	5	"{\"id\": \"79tJyc2F8Dg\", \"original\": null, \"cdate\": 1635904685606, \"pdate\": null, \"odate\":(...TRUNCATED)	{"criticism":3,"example":1,"importance_and_relevance":0,"materials_and_methods":3,"praise":3,"presen(...TRUNCATED)	0.882353	0.629834	0.252519	0.889197	0.076982	0.006844	0.176471	0.058824	0	0.176471	0.176471	0.117647	0.117647	0.058824	{"criticism":0.17647058823529413,"example":0.058823529411764705,"importance_and_relevance":0.0,"mate(...TRUNCATED)	0.882353	iclr2022	openreview	0	0			0	null
zuqcmNVK4c2	Self-Joint Supervised Learning	"Supervised learning is a fundamental framework used to train machine learning systems. A supervised(...TRUNCATED)	"Published as a conference paper at ICLR 2022\nSELF -JOINT SUPERVISED LEARNING\nNavid Kardan†∗, (...TRUNCATED)	Navid Kardan, Mubarak Shah, Mitch Hill	Accept (Poster)	2,022	"{\"id\": \"zuqcmNVK4c2\", \"original\": \"LC1zzwgqdrA\", \"cdate\": 1632875666960, \"pdate\": 16434(...TRUNCATED)	"[Main Review]:\nThe empirical results look convincing, and the Authors' interpretation of the metho(...TRUNCATED)	ICLR.cc/2022/Conference/Paper3122/Reviewer_sxB6	null	4	"{\"id\": \"4l7GAqeKj4I\", \"original\": null, \"cdate\": 1635873986271, \"pdate\": null, \"odate\":(...TRUNCATED)	{"criticism":2,"example":1,"importance_and_relevance":0,"materials_and_methods":14,"praise":1,"prese(...TRUNCATED)	1.136364	-0.142013	1.278377	1.153598	0.167607	0.017235	0.090909	0.045455	0	0.636364	0.045455	0.136364	0.136364	0.045455	{"criticism":0.09090909090909091,"example":0.045454545454545456,"importance_and_relevance":0.0,"mate(...TRUNCATED)	1.136364	iclr2022	openreview	0	0			0	null

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