Probabilistic Entity Representation Model for Reasoning over Knowledge Graphs
Authors: Nurendra Choudhary
Nikhil Rao
Sumeet Katariya
Karthik Subbian
Chandan K. Reddy
Venue: NeurIPS 2021 Poster (2021)
Keywords: gaussian space
probabilistic embeddings
knowledge graphs
query reasoning

Abstract: Logical reasoning over Knowledge Graphs (KGs) is a fundamental technique that can provide an efficient querying mechanism over large and incomplete databases. Current approaches employ spatial geometries such as boxes to learn query representations that encompass the answer entities and model the logical operations of projection and intersection. However, their geometry is restrictive and leads to non-smooth strict boundaries, which further results in ambiguous answer entities. Furthermore, previous works propose transformation tricks to handle unions which results in non-closure and, thus, cannot be chained in a stream. In this paper, we propose a Probabilistic Entity Representation Model (PERM) to encode entities as a Multivariate Gaussian density with mean and covariance parameters to capture its semantic position and smooth decision boundary, respectively. Additionally, we also define the closed logical operations of projection, intersection, and union that can be aggregated using an end-to-end objective function. On the logical query reasoning problem, we demonstrate that the proposed PERM significantly outperforms the state-of-the-art methods on various public benchmark KG datasets on standard evaluation metrics. We also evaluate PERM’s competence on a COVID-19 drug-repurposing case study and show that our proposed work is able to recommend drugs with substantially better F1 than current methods. Finally, we demonstrate the working of our PERM’s query answering process through a low-dimensional visualization of the Gaussian representations.

Probabilistic Entity Representation Model for Reasoning over Knowledge Graphs
Nurendra Choudhary', Nikhil Rao', Sumeet Katariya², Karthik Subbian', Chandan K. Reddy Department of Computer Science, Virginia Tech, Arlington, VA
2 Amazon, Palo Alto, CA
nurendrašvt edu, fnikhilsr, katsumee, ksubbian) com, reddyåcs.vt.c. vt.
Abstract
Logical reasoning over Knowledge Graphs (KGs) is fundamental technique that can provide efficient querying mechanism over large and incomplete databases. Current approaches employ spatial geometries such as boxes to learn query rep- resentations that encompass the answer entities and model the logical operations projection and intersection. However, their geometry is restrictive and leads non-smooth strict boundaries, which further results in ambiguous answer entities. Furthermore, previous works propose transformation tricks handle unions which results in non-closure and, thus, cannot be chained in a stream. In this paper, we propose Probabilistic Entity Representation Model (PERM) to encode entities as Multivariate Gaussian density with mean and covariance parameters to capture its semantic position and smooth decision boundary, respectively. Additionally, we also define the closed logical operations of projection, intersection, and union that can be aggregated using an and-to-end objective function. On the logical query rea- soning problem, we demonstrate that the proposed PERM significantly outperforms the ttate-of-the-ane methods on various public benchmark KG datasets on standard evaluation metrics. We also evaluate PERM's competence repurposing case study and show that our proposed work able to recommend drugs with substantially better Fl than current methods. Finally, we demonstrate the working of our PERM's query answering process through low-dimensional visualization of the Gaussian representations.
COVID-19 drug-
Introduction
Knowledge Graphs (KGs) are structured heterogeneous graphs where information is organized as triplets of entity pair and the relation between them. This organization provides a fluid schema with applications in several domains including e-commerce [[]. web ontologies P B and medical research 4 [ Chain reasoning is fundamental problem in KGs, which involves answering a chain of first-order existential (FOE) queries (translation, intersection, and union) using the KGs" relation paths. A myriad of queries can be answered using such logical formulation (some examples are given in Figure D. Current approaches 6 & in the field rely mapping the entities and relations onto representational latent space such that the FOE queries can be reduced to mathematical operations in order further retrieve the relevant answer entities.
Euclidean vectors ? provide a nice mechanism to encode the semantic position the entities by leveraging their neighborhood relations. They utilize a fixed threshold over the vector query for answer entities (such as k-nearest neighbor search). However, queries differ their breadth. Certain queries would lead greater set of answers than others, c.g., query Canadians will result in higher number of answers than query Canadian Turing Award winners. To capture this query behavior, spatial embeddings
learn border parameter that accounts for broadness of
35th Conference on Neural Information Processing Systems (NcurIPS 2021).

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Q: Who (1) are the Canadian (C) and European (t) Turing (T)
Award (A) viinners (WY?
namess [Wijnnererss gategoryyT
nleibizen(W, Europe) cibizen Canada
Q: Which treatment drugs (T) interact with all proteins (P)
associated with SARS diseases (D)?
category
: 2
associate D.)l
Award
Turing Award
Winners
associate
Treatment
GRIGNTTN
SARS-COVE
Protein
Interacts
Europe
Europeans
Intersection
aesociate
union
BARS-COV2N
Protein 2
(a) Drug Repurposing (DRKG).
(b) Open-domain (FB15K).
Figure Sample FOE queries from different datasets that utilize existential quantification (7), intersection (N), and union (U) operations. The simple operations need to be chained together in an end-to-end objective function retrieve relevant results for complex queries.
queries by controlling the volume of space enclosed by the query representations. However, these spatial embeddings rely form solution to the union operation, e.g., the union of two boxes is not a box. Thus, further FOE operations cannot be applied to the union operation. Additionally, their strict borders lead some ambiguity in the border case scenarios and non-smooth distance function, e.g., a point on the border will have much smaller distance if it is considered be inside the box than if it considered be outside. This challenge also applies to other geometric enclosures such as hyperboloids
complex geometries such boxes 7 which do not have closed
density functions 13 14 15| instead vector points for representation learning. While these approaches utilize the representations for modeling individual entities and relations between them, we aim provide closed form solution to logical queries over KGs using the Gaussian density function which enables chaining the queries together. Another crucial difference in our work is handling a stream of queries. Previous approaches rely on Disjunctive Normal Form (DNF) transformation which requires the entire query input. In our model, every operation is closed in the Gaussian space and, thus, operations of large query can be handled individually and aggregated together for the final answers.
Another line of work includes the of structured geometric regions [12
014
Europeans
Europeans
LeCunn
LeCunn
Bengio
LeCunn
Hinton
-010
Citizen
Canadinn
French
Brithh-Cnnddann
0.09
Fumpewn
0.213
1.343
0.146
1.589
1114
Union
0.146
0.211
1114
Canadians
Bengia
Hinton
Hinton
0.06
Bengio
0.05
Gaussian Minture
Canadan
0.146
1.57
1118
Canadians
0.03
(c) Distance of entitics from query space. For comparability, distances are given relative to entity Bengio's distance to European query.
0.00
(a) Union of box queries.
(b) Gaussian mixture queries.
Figure 2: Results of the query Europeans U Canadians Entities in the darker areas have higher probability of being the answers than lighter areas. We can observe from (c) that the non-smooth borders of box geometry do not encompass the answer Hinton.
To alleviate the drawbacks of operations not being closed under unions and border ambiguities, we propose Probabilistic Entity Representation Model (PERM). PERM models entities mixture of Gaussian densities. Gaussian densities have been previously used in natural language processing [14] and graphs 15 to enable more expressive parameterization of decision boundaries. In case,

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utilize mixture of multivariate Gaussian densities due to their intuitive closed form solution for translation, intersection, and union operations. In addition, they can also enable the use of smooth distance function; Mahalanobis distance [16]. Figure 2 provides an example of such case where the non-smooth boundaries of box query embeddings are not able to capture certain answers. We utilize the mean (1) and co-variance (E) parameters of multivariate Gaussian densities to encode the semantic position and spatial query area of an entity, respectively. The closed form solution for the operations allows us to solve complex queries by chaining them in pipeline. PERM does not need to rely on DNF transformations, since all the outputs are closed in the Gaussian space and complex queries can be consolidated in end-to-end objective function, e.g., in Figure 2b Europeans U Canadi ans is a Gaussian mixture and the single objective is to minimize the distance between the mixture and entity Hinton, whereas in the case of boxes (shown in Figure 2a). we have two independent objectives to minimize the distance from each box in the union query. Summarizing, the contributions of our work is as follows:
1. We develop Probabilistic Entity Representation Model (PERM), a method to reason over KGs using (mixture of) Gaussian densities. Gaussians are able provide closed form solution to intersection and union, and also smooth distance function. This enables us to process chain of complex logical queries in an end-to-end objective function.
2. PERM is able outperform the current state-of-the-art baselines on logical query reasoning over standard benchmark datasets. Additionally, it is also able to provide better drug recommendations for COVID-19.
3. PERM is also interpretable since the Gaussian embeddings can be visualized after each query process to understand the complete query representation.
The rest of the paper is organized as follows: Section 2 presents the current work in the field. In section 5 we present PERM and define its various operations. Section 4 provides the formulation for building the reasoning chains for complex queries. We provide the experimental setup and results in section 5
We conclude our paper in section C and present its broader impact in section 7
2 Related Work
The topic of multi-hop chain reasoning over KGs has gained a lot of attention in recent years [
[8 19 M. These approaches utilize vector spaces to model query representation and retrieve results using& a fixed threshold. While such representations are efficient at encoding semantic information, the fixed thresholds that are typically used in these models do not allow for an expressive (adjustable) boundary and, thus, are not best suited for representing queries. Spatial embeddings
g 20 enhance the simple vector representations by adding a learnable border parameter that controls the spatial area around a query representation. These methods have strict borders that rely on non-smooth distance function that creates ambiguity between border cases. On the other hand, in our model, the variance parameter of the query's Gaussian densities creates soft smoothly increasing borders in terms of the Mahalanobis distance. Additionally, the previous methods do not provide closed form solution for unions which we solve using Gaussian mixture models.
Density-based embeddings have seen recent surge of interest in various domains. Word2Gauss [4] provides a method of learning Gaussian densities for words from their distributional semantic information. In addition, the authors further apply this work to knowledge graphs [33. Another approach [1] aims to learn Gaussian graph representations from their network connections. These methods are, however, focused learning semantic information and do not easily extend to logical queries over knowledge graphs. PERM primarily focuses on learning spatial Gaussian densities for queries, while also capturing the semantic information. To achieve this, we derive closed form solutions to FOE queries.
3 Probabilistic Entity Representation Model for Logical Operators
Knowledge Graphs (KG) G
R are heterogeneous graphs that store entities (E) and relations
(R). Each relation E R is a Boolean function r E x E + (True, False) that indicates if the relation r exists between a pair of entities. Without loss of generality, KGs can also be organized as set of triples (e1, r, e2) C G. defined by the Boolean relation function r(e1, eg). In this work,
3

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part of query and another an entity e; E E as a multivariate Gaussian density function; e; = [N,, Ej), where the learnable parameters je; (mean) and Σ, (covariance) indicate the semantic position and the surrounding query focus on the following three FOE operations: translation (t), intersection (Π), and union (U), The operations are defined as below:
qe (Qa) A (11, 22, ... [k C E 3 01
(1)
gn 2n {ui, U2 ... 3 S E 3 01 Aag n.. Na;
(2)
q. VV : f0), U2, ..., Rx [ 3 Uaz Ua.
(3)
where Q+ (e1, Qu {(11, r1); (2).
and a; a= ra va)
where Qt- In. and qu the translation, intersection, and union queries, respectively; and Vi, Vn, and
Vu are the corresponding results [0]]_ As we notice above, each entity has dual nature; one as being candidate answer to a query. In PERM,
model the query space of
density of the entity, respectively. As a candidate, we only consider the and ignore the Σ, of the entity. We define the distance of candidate entity e2 (N 2,) from a query Gaussian ej (u,, Eg) using the Mahalanobis distance [[ given by:
(us e,) = (1g - n4) n4) _ (M, p4)
Additionally, need define the FOE operations for the proposed Probabilistic Entity Representa- tion Model. visual interpretation of the operations; translation, intersection, and union is shown in Figure The operations are defined follows:
Translation (t), Each entity e E and r E R are encoded as Niees E.) and ModN((rr: E,), respectively, We define the translation query representation of an entity with relation as ge and the distance of resultant entity " E Ve from the query as dy given by:
q: = [( + Mr, (r, + o == 9t)
(5)
Intersection (7). Intuitively, the intersection of two Gaussian densities implies random variable that belongs both the densities. Given that the entity densities are independent of each other, we define the intersection of two entity density functions e1, as On and distance of resultant entity en E Vn from the query as dd given by:
Φn = =N Ee, (M Eez) = NNus: Ez): da = dy((r, 9-)
where, Σg Ej + Σう¹
and μ13 = Ex 1 + Σ 1 12)
E3 33- 1I E
We provide a brief sketch the proof that the intersection of Gaussian density functions is closed operation. A complete proof is provided in Appendix A Let us consider two Gaussian PDFs P(O1) ( Σ1) and P(02) N(2,,, Σ2). Their intersection implies a random variable that distributed as the product, P(0)) The intersection P(0) NNu,, Σz) is derived as follows:
P(0) P(O.).P(02)
log( P(0)) (x: - TT (x m1) + (x 122) TT (x 12) (x - 133) Egg (x Ms) = (a- ((r-M)) m1) T (I m1) + (x - (22) "55 (r v2) Comparing coefficients; 25 = MT + 1 Ms = (5, #1 + E u2)
Union (U). We model the union of multiple entities using Gaussian mixtures. The union of entity density functions given by 61, C22 es...., Cn is defined as gu and the distance of resultant entity au E Vu from the query as dº given by:
au Σ (NN Degl; d3 =
Gidx(ou. Eeg)
where, di exp (Mey: Eecs)
Σ/= eap (N (N(Mess Ee
Φ; E ① are the weights for each Gaussian density in the Gaussian mixture, calculated using the self-attention mechanism over the parameters of the Gaussians the mixture, i.e., Meç E,,, Vi -> n.

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(a) Translation (g+)
(b) Intersection (gn)
(c) Union (qu)
(d) Chain Translation (c+)
(e) Chain Intersection (cn)
(1) Chain Union (cu)
Figure 3: The logical single (top row) and chain operations (bottom row) of translation, intersection, and union in the Gaussian space. The operations are closed and will result in either Gaussian density or a Gaussian mixture. The input operands are given blue and red and the resultant Gaussian density/mixture is depicted in purple. For simplicity, the example is given for univariate Gaussian model, but in our work, we use multivariate Gaussian densities.
Chain Reasoning over Knowledge Graphs
We consider the Gaussian density function (embedding of a single entity) as a special case of Gaussian mixture with single component. This ensures that all the operations defined in Section B are closed under the Gaussian space with an output that is either a single (for translations and intersections) or multi-component Gaussian mixture (for unions). Hence, for chaining the queries, need define the logical operators with a Gaussian density and a Gaussian mixture input. In this section, we define the different operators (depicted in Figure in the case of Gaussian mixture input.
Chain Translation. Let us assume that the input query embedding is an n-component mixture p >i= (u, Mx) and we need to translate it with relation N N(,r, 2,). Intuitively, we would like to translate all the Gaussians in the mixture with the relation. Hence, we model this translation as c and the distance from entities 2 Vi df given by:
C
Φ:N(u + pr. (DT+ + Σマ
(8)
d; o + Mry (Ep エーリ)ー
Chain Intersection. A Gaussian mixture union over individual densities. Based on the dis- n tributive law of sets, an intersection over a Gaussian mixture > Ej) and entity e [pee Σ.) implies the union of the intersection between the entity and each Gaussian density in the mixture. Hence, we derive this intersection
and the distance from entities un E Vn as df;
en UL (M,
Σ
E.)W,, E.)
cn = Σ
N. (Merii Doni)
(10)
1=1
where, Σ-1 --- Ex and Meri Eoni(E Me E-1(u) 14)
di = Σ
ondy(un Deni))
(11)
Chain Union. The union of an entity = NNuee Ed) with Gaussian mixture Σn_ d:W(u Σ,) the addition of the entity to the mixture. Hence, the union cu and the distance from entities au Vu
5

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d can be defined as follows:
J (n Σ,) + (eN(Hee Ec)
(12)
d Bod ((,, (ed Ec))
(13)
Implementation Details. To calculate the weights (d, 4) of the Gaussian mixtures, the popular self-attention mechanism [21] The gradient descent over Mahalanobis distance (Eq. 4 and derivation for the product of Gaussians (Eq. 6 are given by [22) and Appendix A respectively. Another important note is that we do not need to compute E for the operations, but rather we only need compute the E-1 Also, storing the complete Σ requires quadratic memory, i.e., Gaussian density of d variables requires d X d parameters for Σ. So, we only store a decomposed matrix LL Thus, for Gaussian density of d variables our memory requirement is d (r+1) parameters (d for μu and x r for Σー²). For computing the μ4 for intersection, in Eq. we use a linear solver (torch. solve) for faster computation. All our models are implemented and run on four Quadro RTX 8000.
L of :-1:--1
in Pytorch 2
5
Experiments
This section describes the experimental setup used to analyze the performance of PERM on various tasks with focus the following research questions:
1. Does PERM's ' query representations perform better than the state-of-the-ard baselines on the task of logical reasoning
standard benchmark knowledge graphs?
2. What is the role of individual components in PERM's ' overall performance gain?
3. Is PERM able recommend better therapeutic drugs for COVID-19 from drug re-purposing graph data compared the current baselines?
Are we able to visualize the operations on PERM's query representations in the latent space?
5.1 Datasets and Baselines
We utilize the following standard benchmark datasets to compare PERM's performance on the task of reasoning over KGs:
FB15K-237 24 is comprised of the 149,689 relation triples and textual mentions of Freebase
entity pairs. All the simply invertible relations are removed.
NELL.995 consists of 107,98? triples obtained from the 995th iteration of the Never-Ending Language Learning (NELL) system.
DBPedia is subset of the Wikipedia snapshot that consists of multi-level hierarchical taxonomy over 240,942 articles.
DRKG 20 (Drug Re-purposing Knowledge Graph) is used to evaluate the performance of our model on both the logical reasoning and drug recommendation tasks.
Table 1: Dataset statistics including the number unique entities, relations, and edges, along with the splits of dataset triples used in the experiments.
Dataset Entities
Relations
Edges
Training
Validation
Test
FB15k-237
14,505
237
310,079
272,115
17,526
20,438
NELL995
63,361
200
142,804
114,213
14,324
14,267
DBPedia
34,575
3
240,942
168,659
24,095
48,188
DRKG
97,238
107
5,874,271
4,111,989
587,428
1,174,854
Implementation code: cade/h/appes//ghhhhh.c.oo/ttttie GaABR---GuuiinnKGG
https://ww/ http://wwe...oo/ooono//ettaaaaaa

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More detailed statistics of these datasets are provided in Table ! For our experiments, we select the following baselines based on (i) their performance the logical reasoning task and (ii) their ability to extend to all FOE query combinations.
Graph Query Embedding (GQE) embeds entities and relations as vector and utilizes TransE [17 to learn the query embeddings. The distance of the answer entities is calculated using Ll-norm.
Query2Box (Q2B) embeds entities and relations axis aligned hyper-rectangles boxes and utilize FOE queries to learn query representations, The distance of answer entities is given by weighted combination of the answer's distance from the center and the border of the query box.
Beta Query Embedding (BQE) [1 utilizes beta distribution learn query representations from FOE queries with a novel addition of negation queries. The distance is calculated as the dimension-wise KL. divergence between the answer entity and the query beta embedding.
Complex Query Decomposition (CQD) 10 answers complex queries by reducing them to simpler sub-queries and aggregating the resultant scores with t-norms.
Some of the other baselines intuitively extended to handle all FOE queries, and hence, we did not include them in our study.
2 focus solely the multi-hop problem. They could not be
Table 2: Performance comparison PERM against the baselines to study the efficacy of the query representations. The columns present the different query structures and the overall average performance. The last row presents the Average Relative Improvement (%) of PERM compared to CQD over all datasets across different query types. Best results for each dataset are shown in bold. The MRR results for experiments are given in Appendix C
HITS@1
@3
Dataset
Model
3t
20
3n
2U
nt
tn
Ut
Avg
FB15K-237
GQE
.404
.214
.147
266
.390
.164
.087
.162
.155
.221
BQE
.455
.122
.102
.32
.459
.141
.244
.124
.101
.218
Q2B
.467
.440
.186
.324
.453
.239
.050
.108
.193
.251
CQD
.512
.288
.221
.322
.457
.284
.129
.249
.121
.290
PERM
.520
286
.216
.361
.490
.305
.128
.212
.239
.306
NELL995
GQE
.417
.231
.203
.388
.454
.200
.081
.188
.139
.248
BQE
.711
.156
.132
.438
.540
.153
.250
.160
.091
.292
Q2B
.555
266
2333
343
.480
.369
.132
.212
.163
.306
CQD
.667
.350
.288
.410
.529
.531
.171
.277
.156
.375
PERM
.581
.266
.243
.352
.508
.460
.143
.195
.200
.328
DBPedia
GQE
.673
.006
N.A.
.833
.879
.402
.160
.668
0.00
.458
BQE
881
.007
N.A.
1.00
1.00
.384
.435
.590
0.00
.655
Q2B
.832
.0073
N.A.
1.00
1.00
.649
.224
.856
0.00
.571
CQD
.870
.0073
N.A.
1.00
1.00
.673
.218
.787
0.00
.569
PERM
.950
.0073
N.A.
1.00
1.00
.782
.232
.952
0.00
.615
DRKG
GQE
.200
.188
.153
270
.409
.181
101
.186
.174
.235
BQE
.554
.141
.123
347
-512
.185
.281
.173
.124
.271
Q2B
.499
.263
.199
.37
.489
.284
.068
.134
.235
.279
CQD
.554
.323
.238
369
.495
.341
.184
.310
.150
.329
PERM
.565
.322
.236
.387
.540
.376
.190
.273
.297
.354
PERM Q2B (%)
10.9
12.3
13.0
7.20
6.10
26.3
84.2
50.8
24.3
15.9
PERM vs CQD (%)
3.80
-0.9
-2.4
2.00
5.80
9.50
1.80
-5.5
93.0
6.2
5.2 (RQ1) Reasoning over KGs
To evaluate the efficacy of PERM's query representations, we compare against the baselines on different FOE query types; (i) Single Operator: 1t, 2t, 3t, 2n, 3n, 2U and (ii) Compound Queries:
nt, tn, Ut. We follow the standard evaluation protocol 7 1 and utilize the three splits a KG
DDBPedia has an extremely large number of resultant grand-childeed leaves (z= 10 per grand-parent for the 2t task and, thus, notice poor performance 2t task across all the evaluation models.
7

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for training Gtrain, validation Ivalid, and evaluation Gtest (details in Table I. The models are trained on Girain with validation on Inald- The final evaluation metrics for comparison are calculated on Grest- For the baselines, we calculate the relevance of the answer entities to the queries based on the distance measures proposed in their respective papers. PHRM, the distance of the answer entity from the query Gaussian density is computed according to the measures discussed in Sections $ and
We use the evaluation metrics HITS@K K and MRR to compare the ranked set results obtained from different models. Given the ground truth E and model outputs {ei, ...,... Cn E E, the metrics calculated follows:
[1 E E
HITS@K E
f(ex);
f(ex)
K
10 else
k=1
MRR 1
Σ ==1
f(e)) = i, i E
00, else
From the results provided in Table 2 we observe that PERM, is able to outperform all the current state-of-the-ard approaches, on an average across all FOE queries by 6.2%. Specifically, we see consistent improvement for union queries; 9.5% and 93% in the case of 2U and Ut., respectively. Comparing the models based on only geometries, we notice the clear efficacy PERM query representations with average improvement of 37.9%, 15.9%, and 37.3% over vectors (GQE), boxes (Q2B), and beta distribution (BQE), respectively. Given these improvements and the ability to handle compound queries in an end-to-end manner, we conclude that Gaussian distributions are better at learning query representations for FOE reasoning over KGs. Additionally, we provide PERM's results sample queries from different datasets in Table
Table 3: Qualitative results of PERM on samples from different datasets. Results given in green and red indicate correct and incorrect prediction, respectively.
Query
Results
Who are European and Canadian Turing awards winners?
Jeffrey Hinton, Yoshua Bengio, Andrew Yao
Which Actors and Football Players also became
Amold Schwarzenegger, Heath Shuler, Frank White
Governors?
Which treatment drugs interact with all proteins associated with SARS diseases?
Ribavirin, Dexamethasone, Hydroxychloro- quine
5.3 (RQ2) Ablation Study
In this section, we evaluate the need for different components and their effects on the overall performance of our model. First, we look at the contribution of utilizing different types of queries to the performance (i) only It queries, (ii) only translation (It,2t,3t) queries and (iii) only single operator queries (1t,2,,33,2n,3)) 3,2, Furthermore, we look the need for attentive aggregation the case of union of Gaussian mixtures. We test other methods of aggregation; (i) vanilla averaging and (ii) MLP [28
our model. For this, we train our model on different subsets of queries;
Table Ablation study results. Performance comparison of PERM (final) against different variants of our model. 1t, translation and single utilize the 1-hop queries, all translation queries and all single operator queries, respectively. The average and MLP variants utilize vanilla averaging and MLP for aggregation in union queries. The metrics reported here are average over all the datasets. Finer evaluation with results for each dataset is given in Appendix D Best results are given in bold.
HITS@3
Model Variants
1t
3t
2n
3n
20
ntt
Ut
Avg
PERM-1t
.649
.141
.128
.400
.466
.477
.095
.257
.102
.303
PERM-translatin translation
.649
.182
.179
.463
.555
.479
.128
.308
.143
.341
PERM-sings -single
.652
.225
.228
524
.632
.475
.167
.398
.181
.387
FFRM-avvragee
.628
.222
.224
.524
.624
.444
.158
.387
.180
.377
PERM-MIP
.642
.225
.228
.26
.631
.462
.166
.400
183
.385
PERM (final)
.654
.225
.232
.255
.635
.481
.170
.408
.184
.390

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From Table 4 we notice that utilizing only It queries significantly reduces the performance of our model by 22.3% and even increasing the scope all translation queries still lower in performance by 12.5% for this case. However, we notice that training on all single operator queries results in comparable performance to the final PERM model. But, given the better overall performance, we utilize all the queries in our final model. For union aggregation, we observe that attention has clear advantage and both vanilla averaging and MLP lead to lower performance by 3.33% and 1.28%, respectively, Thus, we adopt self-attention in our final model.
5.4 (RQ3) Case Study: Drug Recommendation
In this experiment, we utilize the expressive power of PERM's 's query representations to recommend therapeutic drugs for COVID-19 from the DRKG dataset. Drugs in the dataset are already approved for other discases and the aim is to utilize the trug-protein-diseas networks and employ them towards treating COVID-19. This can potentially reduce both the drug development time and cost [29]. For this experiment,
utilize the treatment relation in DRKG and retrieve drugs D D treats, X, where is a set of SARS diseases related to the COVID-19 virus. Given that we only need these limited set of entity types (only SARS diseases and drugs) and relation types (only treatments), only consider the DRKG subgraph that contains this necessary set of entities and relations for learning the representations. We compare the recommendations of different models against set of actual candidates currently in trials for COVID-19. We use the top- lop10 recommendations with the evaluation metries of precision, recall, and Fl-score for comparison.
Table 5: Performance comparison of various models on the COVID-19 drug recommendation problem using precision (P), recall (R), and Fl-score (F1) metrics. The top three drugs recommended by the models are given in the last column. The recommendations given in green and red indicate correct and incorrect predictions, respectively. The last two rows provide the average relative improvement of PERM compared to the state-of the art baselines Q2B and COD.
Model
P@ 10
R 10
F1
Top Recommended Drugs Piclidenoson, Ibuprofen, Chloroquine Ossetamii,,slttmiiiii Ruxolitinib
GQE
.119
.174
.141
BQE
.159
200
.177
Q2B
.194
.255
.221
Ribavirin, Dexamethasone, Deferoxamine Ribavirin, Dexamethasone, Tofacitinib Ribavirin, Dexamethasone, Hydroxychloroquine
CQD
.209
.260
.232
PERM
.217
.269
.251
PERM Q2B (%)
11.9
5.5
13.6
PERM vs CQD (%)
3,8
3.5
8.2
We can observe from Table that that PERM is able to provide the best drug recommendations, across all evaluation metrics. Our model is able to outperform the current methods by atleast 3.8%, 3.5%, and 8.2% in precision, recall, and F1, respectively. Also, the top recommended drugs by our PERM are more inline with the current drug development candidates, thus, showing the better performance of model's query representations.
5.5 (RQ4) Visualization of the Gaussian Representations
To visualize the entity and query in the latent space, extract representative entity samples from the FB15K-237 dataset and present them a 2-dimensional space for better comprehension.
Figure depicts the different entities and the mechanism through which PERM narrows down to the particular answer set. Notice that, we are able to perform an intersection after union operation due to the closed form nature of our operations. This is currently not possible in state-of-the ff-the-art art baseline methods. Additionally, it should be noted that, unions widen the query space and intersections narrow them down (as expected). Furthermore, the variance parameter acts as control over the spatial area that an entity should cover and more general entities such as Turing Award and Europe occupy a larger area than their respective sub-categories, namely, winners and Europeans.

---

Europeans
Turing Award Winners
Canadian (C) European Turing (A) (W)?
Awards
Turing
winners
Europe
Europeans
Intersection
Final Query Representation
Union
Canadiane
Canada
Canadians
(a) Query processing in PERM. This figure depicts univariate version of the entity Gaussian embeddings for better visualization of the process. The same property, however, generalizes over an increased number of dimensions, i.c., multivariate case.
(b) Bivariate version of the final query space, given in grayscale with darker colors representing a higher probability of answers.
Figure 4: An illustration of the flow for sample complex query in the representational space. We note that intersection after union is possible in our PERM model because the operations are closed in Gaussian distributions and this is not possible in current methods including BQE, Q2B, and CQD.
Conclusion
In this paper, we present Probabilistic Entity Representation Model (PERM), a model learn query representations for chain reasoning over knowledge graphs. We show the representational power of PERM by defining closed form solutions to FOE queries and their chains. Additionally, we also demonstrate its superior performance compared to its state-of-the-ard counterparts on the problems reasoning over KGs and drug recommendation for COVID- 19 from the DRKG dataset. Furthermore, we exhibit its interpretability by depicting the representational space through sample query processing pipeline.
Broader Impact
PERM is the first method that models an individual entity in knowledge graphs using Gaussian density function, making it possible to solve FOE queries using a closed form solution. This enables its application in domains that require chain reasoning. The main idea of the proposed solution can also be extended to any domain that can encode its basic units as Gaussians and extend the units through FOE queries, e.g., in topic modeling, topies can be encoded Gaussians and documents as union of topics.
However, PERM depends the integrity of the knowledge graph used for training. Any malicious attacks/errors NO 3II that lead to incorrect relations could, further, lead to incorrect results and affect the confidence of our model. Furthermore, due the connected nature of complex queries, this attack could propagate and affect a larger set of queries. Such incorrect results would be problematic in sensitive areas of research such drug recommendations and cybersecurity and, thus, it is necessary to maintain the integrity of training data before learning representations and querying with PERM.
Acknowledgments
Funding: This work was supported in part by the US National Science Foundation grant IIS- 1838730 and Amazon AWS cloud computing credits for research.
Competing Interests: All the authors had some form of affiliation with Amazon in the past 36 months.
10

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12


Reviews:
[{"review_id": "hwdhEWVmRfY", "reviewer": "Reviewer_dcyX", "summary": "For the multi-hop logical reasoning in the knowledge graph, this paper presents a probabilistic entity representation model. The proposed model encodes entities in knowledge graphs using a multivariate Gaussian density function. Then, this paper defines logical operations such as translation, intersection, and intersection and chained operations using Gaussian properties with the proposed model.", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "The proposed method that models an entity in knowledge graphs using the Gaussian density function seems reasonable. Furthermore, the logical operations and the chain reasoning are properly defined.\n\nBut, this paper leaves some negative or uncertain points.\nAs this paper mentioned, some previous studies have modeled both entities and queries by probabilistic distributions such as hyperboloid embeddings [8] and beta embeddings [11]. So, there is one question about the motivation of the proposed method. What is the main advantage of using the Gaussian density function? It would be better to include some analyses and/or discussions about the comparison with the methods using another probabilistic distributions instead of comparing box geometric approach.\n\nThe proposed method only supports existential positive first-order queries. That is, this paper cannot consider negation logical operation. Compared to the previous study (Beta embeddings [11]) that deals with the negation logical operation, the proposed method lacks the coverage of first-order logic.\n\nThe main concerns of this paper involve the experimental settings and results. First, the FB15k-237 and NELL995 datasets used in this paper are different from the ones used in previous studies [7, 10]. For example, according to Section C in the appendix, the number of relations and edges of FB15k-237 are 474 and 149,689, respectively. However, in [7, 10] papers, those are 237 and 272,115. The reviewer was wondering why this paper does not use the same dataset used in the previous studies [7, 10]. The reviewer speculates that the dataset used in this paper is the one used in [8]. However, the dataset of [8] is not the same as the one of [7, 10].\n\nSecond, the performances in Table 1 are quite strange. It looks like that the performances of BQE and CQD are borrowed from each paper, respectively. Since the dataset is different, their performances cannot be borrowed directly. For example, the BQE paper [10] provides the performances of HITS@1 and MRR, not the one of HITS@3. However, this paper uses MRR as HITS@3. In addition, the performance of BetaE in Table 1 and the one in Table 5 (in appendix) is the same. For these reasons, the reviewer cannot trust the experimental results.\n\nIt would be better to mention/include a training objective of the proposed model in detail. \n\nIn Section 5.5, it would be better to include entity distribution as well as query distribution.\n\n==\n\n* Update after author responses *\n\nI want to thank the authors for the additional information provided in a short period. The authors have provided experimental results on standard dataset. Since the concerns about the experiments were addressed, I raised my score.", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "It is properly addressed.", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "24"}, "extra_text": {}}, {"review_id": "IFDokcJ_V3", "reviewer": "Reviewer_y6MJ", "summary": "This paper proposed a new method for knowledge graph reasoning: modeling the entity and relation with a probabilistic distribution rather than a single vector. Then the operations can be regarded as a function of probability distribution transformation. With an assumption of Gaussian distribution families, the entire process can be considered in a tractable manner. Empirical results on multiple datasets show that the method outperforms previous SOTA systems. The general idea is novel and very interesting. The experimental results are convincing. The writing is clear and well-structured. I like the idea a lot.\n\n", "questions": "", "limitations": "", "rating": 7, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Pros:\n\n1. A novel and interesting idea for modeling the entities and relations with a random variable with gaussian probabilistic distributions.\n2. The designed operations are cleverly designed and seem reasonable to me.\n3. The empirical results are convincing\n4. The analysis is very interesting. I like Sec 5.5 a lot.\n\nCons: \n\nIf my understanding is correct, the logical reasoning step (Union->Intersection-> … ) must be given in both training and testing. If so, I am not sure it is okay to claim that the proposed approach is “to solve FOE queries in a closed form solution”. My intuition is that inferring logical steps might be the most difficult problem.", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "yes", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": ["I don’t know"], "time_spent_reviewing": "3"}, "extra_text": {}}, {"review_id": "7UsJwPecvZV", "reviewer": "Reviewer_MHQ9", "summary": "\nIn this paper authors proposed a method to solve logical reasoning over knowledge graphs. They proposed the use of gaussian density function instead of previous methods like union of boxes. This shows good results in the experiments and also good visualization of the distribution.", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Originality: \n\nThe work is a novel addition of gaussian density function into the existing logical reasoning over knowledge graphs work. I like the intuitive explanation from the authors on how it gives better expressibility for queries over the knowledge graph.\n\nQuality: \n\nThe submission seems technically sound. The motivation is well explained with examples. And the result also mostly supports the hypothesis.\n\nAlthough the final average is better for PERM vs CQD, but in some queries like 2t, 3t, t\\intersection etc CQD seems to outperform PERM. Some analysis into these particular queries may shed more light about why this happens. It might give some query specific intuition and may help improve the work more.\n\nI also have questions regarding how many queries were used for each type. And how were they averaged. Is the average weighted on number of queries? If some query class dominates the average. Because we are showing efficacy based on the average of all queries, I think an analysis of how the queries were chosen is important.\n\nDrug recommendation experiment seems interesting. How was the ground truth set was constructed for this? Is it all the drugs that currently are in trial? It would be good to know how good each drug works, but that data is probably not available at this point. But this shows good potential for drug recommendation.\n\nI liked the visualization of the gaussian representation. It gives some intuitive sense to the work.\n\nClarity: \n\nThe submission is clearly written and seems well organized and easy to read. \n\nSignificance: \n\nI think this is a very important problem to solve in this area. It should be impactful work for researchers in this domain.\n\n", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "did not see this section", "review_text": "", "extra_scores": {"ethical_concerns": "no concerns", "needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "1"}, "extra_text": {}}, {"review_id": "-As4QL7RlQ", "reviewer": "Reviewer_LbhD", "summary": "The paper presents Probabilistic Entity Representation Model (PERM), a technique that models entities in a Knowledge Graph (KG) through a mixture of Gaussian densities.", "questions": "", "limitations": "", "rating": 7, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Strengths:\n- The paper is well written and easy to follow.\n- The proposed technique unlocks not-trivial gains with respect to the state-of-the-art on several datasets.\n- Although gaussian densities are not a new concept, and have been successfully applied to domains such as NLP, their application to chain reasoning over KG can be considered novel.\n- The code to reproduce the experiments is available and will be made public.\n\nWeakness:\n- The numbers for CQD in Table 1 seems not to match those reported in their paper. For instance, for NELL995 1t should be 0.667 instead of  0.567, and so on. Please double check.\n- I would have appreciated a discussion section, potentially highlighting some of the limitations of the proposed approach. Interesting questions that comes to mind are: (1) Is PERM robust against missing information in a KG? To what extent? (2) Is the proposed solution differentiable? Can it be trained end2end in the context of a broader framework?\n\nPresentation comments:\n- The first sentence in the abstract is a bit hard to parse. Why does logical reasoning over KG limit its application on large DB? Please rephrase or add a sentence to elaborate.\n- The fact that KG's organisation provides a *fluid*-schema can be disputed (line 21), it is extremely static from my point of view. Please elaborate on the fluid adjective or rephrase. \n- Figure 1 is very hard to read on a A4 paper, you should make the font larger.\n- Most of the Figures are not colour-blind friendly (in particular, Figure 3, Table 3 and Figure 4). Please make them so.\n- Examples of good and bad PERM predictions could be useful.\n", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "the authors adequately addressed the limitations and potential negative societal impact of their work", "review_text": "", "extra_scores": {"ethical_concerns": "none", "needs_ethics_review": "No", "ethics_review_area": ["I don’t know"], "time_spent_reviewing": "2"}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper presents a probabilistic treatment of KG entities, and introduces a logic over such probabilistic entity representations which supports multi-step reasoning about relationships in graphs. It shows the strengths of this approach on a number of graph tasks, and is clearly written. There is a fairly unanimous consensus from the reviewers that this is a strong paper, and worth publishing.

From a cursory read through, given the focus on logical reasoning on graphs, I was surprised not to see mentions of [1,2] which build on [3] (I should indicate that these are not my works, but just that I am familiar with this set of approaches) in at least related work, if not direct comparison (which would be possible on some datasets). I see the approach presented here as being very different in spirit and execution, so while this absence is surprising I don't think it is damning. I just indicate this remark FYI, and did not take it into consideration when making a recommendation for this paper.

I am happy to go with the reviewer consensus and recommend acceptance for this paper at NeurIPS.

[1] Minervini, Pasquale, et al. "Differentiable reasoning on large knowledge bases and natural language." Proceedings of the AAAI conference on artificial intelligence. Vol. 34. No. 04. 2020.

[2] Minervini, Pasquale, et al. "Learning reasoning strategies in end-to-end differentiable proving." International Conference on Machine Learning. PMLR, 2020.

[3] Rocktäschel, Tim, and Sebastian Riedel. "End-to-end differentiable proving." arXiv preprint arXiv:1705.11040 (2017)
Decision: Accept (Poster)