C UBE R EP : L EARNING R ELATIONS BETWEEN DIFFERENT VIEWS OF DATA
Anonymous authors
Paper under double-blind review
A BSTRACT
Multi-view learning tasks typically seek an aggregate synthesis of multiple views or perspectives of a single data set.
The current approach assumes that there is an ambient space X in which the views are images of X under certain functions and attempts to learn these functions via a neural network.
Unfortunately, such an approach neglects to consider the geometry of the ambient space.
Hierarchically hyperbolic spaces (HHSes) do, however, provide a natural multi-view arrangement of data; they provide geometric tools for the assembly of different views of a single data set into a coherent global space, a CAT(0) cube complex .
In this work, we provide the ﬁrst step toward theoretically justiﬁable methods for learning embeddings of multi-view data sets into CAT(0) cube complexes.
We present an algorithm which, given a ﬁnite set of ﬁnite metric spaces (views) on a ﬁnite set of points (the objects), produces the key components of an HHS structure.
From this structure, we can produce a CAT(0) cube complex that encodes the hyperbolic geometry in the data while simultaneously allowing for Euclidean features given by the detected relations among the views.
I NTRODUCTION
There are many data analysis tasks for which we have multiple views of the same aggregate data set and for which we must synthesize the different views.
These views might be images of objects from different angles or in different lighting (Geusebroek et al., 2005), or the translations of a single text into multiple languages (Christodoulopoulos & Steedman, 2015).
In many cases, these views encode similar information but they also have complementary information that is important to capture.
Aggregating all of the information into a coherent global framework is crucial for obtaining the best performance on downstream tasks.
The predominant current approach to multi-view learning is to assume that there is an ambient space
X such that the views are images of X under certain functions.
Such an approach, while successful, has drawbacks.
Notably, the choice of the geometry of the space in which we represent our aggregate data set is currently made without theoretical considerations.
These methods (Xu et al.
; Sun et al.
; Li et al.
; Zhao et al.
; Wu et al.
; Han et al.
(2021)) also assume the maps from the ambient space to the views are obtained via a neural network, which is used simply as a blackbox and thus prevents mathematical insight and explicability.
Multi-view arrangements of data naturally arise in geometric group theory in the context of hierarchically hyperbolic spaces (HHSes) (Behrstock et al.
This hierarchical approach builds on work in several areas of low dimensional topology, including mapping class groups (Masur &
Minsky (2000), Teichm¨uller spaces (Brock (2003); Raﬁ (2014); Durham (2016)), and hyperbolic 3-manifolds (Minsky (2010); Brock et al.
HHSes provide geometric tools for the assembly of different views of a single data set into a coherent global space, a CAT(0) cube complex .
CAT(0) cube complexes are combinatorial objects that are capable of blending a mixture of both hyperbolic (e.g., tree-like) and Euclidean geometries (Sageev (2014)).
We note that previous work such as Billera et al.
; John (2017) exploited the cubical structure of the space of phylogenetic trees to improve the reconstruction of phylogeny from genetic data.
In this work, we take the ﬁrst step toward theoretically justiﬁable methods for learning embeddings of multi-view data sets into CAT(0) cube complexes.
Speciﬁcally, given a ﬁnite collection of objects x 1 , . . .
, x n , each occurring as points in a ﬁnite collection of views V 1 , . .
. , V K , one can produce a ﬁnite collection T 1 , . .
, T K of ﬁnite trees with leaves the x i .
One method for learning these trees is T REE R EP (Sonthalia & Gilbert, 2020).
We develop algorithms which learn certain relations among the views and relative projection data, from which one can produce an HHS structure on { x 1 , . . .
, x n } .
From this HHS structure, we can produce a CAT(0) cube complex that encodes the hyperbolic geometry in the T i while simultaneously allowing for Euclidean features given by the detected relations among the V
i (Behrstock et al.
In this paper, our main purpose is to present an algorithm which, given a ﬁnite set of ﬁnite metric spaces (views) on a ﬁnite set of points (the objects), produces the key components of an HHS structure.
B ACKGROUND
[Figure]
Fix n objects x 1 , . . .
, x n and K metric spaces V 1 , . . .
, V K , with each x i representing a point in each
V j .
We think of the x i as being global objects and their representatives in each view V j as a sort of projection.
For example, take a set of points { x i } ni =1 ⊂ R K and V 1 , . .
, V k the K coordinate axes.
Then, the representatives of the x i in the j th copy of R are simply the j th coordinate of each x i .
An HHS is deﬁned axiomatically by relations amongst its constituent metric spaces (Behrstock et al.
We begin with an ambient metric space X with a (usually inﬁnite) collection of
Gromov hyperbolic geodesic metric spaces V and a family of projections π V : X → V for each
V ∈ V .
Given these spaces, we can always construct an ambient product space (cid:81) V V with the map
X → (cid:81) V V given by x (cid:55)→ ( π V ( x )) V ∈V , whose image is controlled by certain relations among the elements of V .
Two spaces V, W can be orthogonal V ⊥ W , transverse V (cid:116) W , or nested V (cid:118) W , though we will not consider nesting here.
Orthogonality and transversality are symmetric relations that generate a family of inequalities which constrain the coordinates of the image X
→ (cid:81) V V .
The orthogonality relation V ⊥ W imposes no constraints; i.e. the map X
→ V × W given by x (cid:55)→ ( π V ( x ) , π W ( x )) is surjective (that is, the map maps X onto V × W ).
On the other hand, transversality V (cid:116) W provides certain consistency inequalities (CI).
These are deﬁned in terms of relative projections ρ VW : V → W, ρ WV : W → V for
V (cid:116) W , whose images have bounded diameter in W and V , respectively, and thresholds θ V >for each V ∈ V .
(CI)
For any x, y ∈ X , if D V ( π V ( x ) , π V ( y )) > θ V and D W ( π W ( x ) , π W ( y ) >
θ W , then up to switching the roles of x, y , either d V ( x, ρ WV ) < θ V or d W ( y, ρ VW ) <
θ W .
Note that this is an exclusive or, that is, exactly one of the d V ( x, ρ WV ) <
θ V or d W ( y, ρ VW ) <
θ W is true.
For each pair x, y ∈ X , these consistency inequalities can be converted into a partial order < x,y on the set
[Equation]
The partial orders across the pairs are further constrained by the following bounded geodesic image ( BGI ) property:
(BGI)
If x, y ∈ X and V, W ∈ L ( x, y ) with V (cid:116) W , then any geodesic [ x, y ] V in V between x, y satisﬁes d V ([ x, y ] V , ρ WV ) <
θ V .
The idea is that if we have V i < x,y V j , then when moving efﬁciently from x to y in the ambient space X , we ﬁrst have to move in V i and then in V j .
Let us look at the examples in Figure 1 to understand these deﬁnitions.
First, consider the 4 -cycle as X in (a).
Let us assume that we have two views V (1 , 2) , V (1 , 4) given by the projections onto the edges (1 , 2) and (1 , 4) .
Now we can see that the map X → V (1 , 2) × V (1 , 4) is surjective (i.e., onto) and, hence, the views are independent.
On the other hand, consider the cross in part (b), with views given by the edges V (1 , 2) and V (1 , 3) .
Then we see that X → V (1 , 2) × V (1 , 3) is not surjective.
Further, when going from node 2 to node 3 , we ﬁrst have to traverse the edge (2 , 1) and then the edge (1 , 3) .
Thus, we see that V (1 , 2) < 2 , 3 V (1 , 3) and that the views are transverse.
We brieﬂy remark on the CAT(0) cube complexes produced from the HHS structure on these synthetic examples.
In the case when the graph is a tree, the resulting CAT(0) space is again the tree.
And when the graph contains cycles, we get a simplicial complex whose 1-skeleton is the underlying graph, with higher dimensional cells corresponding to pairwise-orthogonal collections of views.
A LGORITHM
For each pair of points x s , x t , we want to learn the relations between the views of the data as we travel from x s to x t in the ambient space X .
To do this, we need to learn the two quantities:
The thresholds θ V k for V k ∈ V . 2.
A directed acyclic graph G st that represents the partial ordering given by the (BGI) property.
Given threshold values θ V k , and a pair of points x s and x t , we add the directed edge ( V i , V j ) in G st if there exists two points x r 1 and x r 2 such that
[Equation]
and for every new pair of points x s (cid:48) , x t (cid:48) , we have that
[Equation]
Here we assume that
ρ V
i V j
( V i ) =
x
r 2 and ρ V j V i ( V j ) =
x r 1 .
Hence satisfying Equation 1 implies that
Property (CI) is satisﬁed and satisfying Equations 2 that Property (BGI) is satisﬁed.
For a given set of thresholds, this directed graph could have cycles.
Hence we want to measure how far the directed graph is from being acyclic.
To do so, we compute L ( G ) =
( Tr (exp( G ) − K )) 2 .
From Wei et al.
, we know that L ( G ) is zero if and only if G is a directed acyclic graph.
Thus, our overall loss function is
[Equation]
, θ V K
[Equation]
:= (cid:88) s (cid:54) [Equation] L ( G st ) . One idea would be optimize for the threshold values θ V i via gradient descent, but due to the combinatorial nature of the objects the gradient of the loss function with respect to the thresholds will always be 0 .
Instead we will do a random walk on a mesh in R K as follows.
Given a view V i , let d ( i ) 1 < . . .
< d ( i ) M
i be the unique distances in d V i .
Now for any θ V i in the interval ( d ( i ) a , d ( i ) b ) , the constructed graphs are the same.
Hence we can pick M i − 1 values ( τ ( i ) l ) l =1 ,...
M i − 1 as the midpoints of these intervals, as well as one point τ ( i ) M i bigger than d ( i ) M i and one point τ ( i ) 0 smaller than d ( i ) 1 .
Thus, we will think of our thresholds as living in the folllowing mesh.
[Equation]
For the random walk, given Θ old , we randomly pick a view V k and then update θ V k to a uniformly random value in { τ ( k ) 0 , . . .
, τ ( k ) M k } to get Θ new .
Then for both sets of thresholds we compute the loss values L (Θ old ) , L (Θ new ) .
If L (Θ new ) <
L (Θ old ) , we move to the new set of thresholds, otherwise we move to the new state with probability (cid:18) L
[Equation]
Θ old )
L (Θ new ) (cid:19).
Thus, our algorithm is a Markov chain
Monte Carlo method.
E XPERIMENTS
We test our algorithms on synthetic data to demonstrate that we ﬁnd the correct relations.
We ﬁrst look at simple undirected graphs as the ambient space.
The nodes of the graph are our data points and our views are the projections onto the edges of the graph.
The idea is to learn an HHS structure on this information which will allow us to rebuild the graph.
For the ﬁrst test, we start with two very simple examples seen in Figure 1.
We run our method a 1000 times to see which relations persistently appear.
For the 4 -cycle, we never get any relations.
Thus, we always get that the views are independent.
On the other hand, for the cross, we get that every view is related to every other view.
Thus, we see that our method works in this setting.
In the ﬁrst generalization, we can look at graphs that are even cycles.
For these graphs we should never have any relations between the views.
We can also look at graphs that are trees.
For these graphs, we should have that every pair of views is related.
Running our method on these graphs provides relations that are consistent with our expectations.
Finally, we look at graphs that have both cycles and tree like parts.
Speciﬁcally, we look at bipartite graphs for which no cycle has a chord.
Examples of such graphs can be seen in Figure 2.
Let us look at the graph on the left of Figure as an example.
For this graph, we expect that the edges (views) in the cycle, (1 , 3) , (3 , 7) , (7 , 5) , (5 , 1) , are independent of each other and that all other pairs of views are transverse.
This is exactly the set of relations that we get when we run our method.
Thus, we see that on these simple instances our method learns the correct relations between the views.
We note that the relations and relative projections produced by our algorithms are sufﬁcient for rebuilding the graph.
While this is possible to do directly from the initial edge and projection data, we believe that these synthetic experiments are a good proof of concept.
F UTURE W ORK AND C ONCLUSION
We considered using multi-view metric data to reconstruct an ambient space that is consistent with the views of the data.
We developed an algorithm that uses ideas from geometric group theory to learn relations between the views.
Our algorithm is an MCMC algorithm that use DAG learning inspired loss functions.
Using our algorithm, we show that for certain types of undirected graphs, we exactly recover the correct relations between the views.
We have shown so far that our method accurately recovers the relations between the views.
There is, however, more work to be done: (1) develop an algorithm that converts a learned HHS structure into the corresponding CAT(0) space, (2) test the algorithm on other types of synthetic data, as well as real world data, and (3) speed up the learning algorithm.