Metric Recovery from Unweighted k-NN Graphs

1 KYOTO UNIVERSITY
KYOTO UNIVERSITY
Metric Recovery from Unweighted k-NN Graphs
Ryoma Sato

2 / 45 KYOTO UNIVERSITY
I introduce my favorite topic and its applications

Metric recovery from unweighted k-NN graphs is my
recent favorite technique.
I like this technique because
The scope of applications is broad, and
The results are simple but non-trivial.

I first introduce this problem.

I then introduce my recent projects that used this technique.
- Towards Principled User-side Recommender
Systems (CIKM 2022)
- Graph Neural Networks can Recover the Hidden
Features Solely from the Graph Structure (ICML 2023)

Metric Recovery from Unweighted k-NN Graphs
Morteza Alamgir, Ulrike von Luxburg. Shortest path distance in random k-nearest neighbor graphs. ICML 2012.
Tatsunori Hashimoto, Yi Sun, Tommi Jaakkola. Metric recovery from directed unweighted graphs. AISTATS 2015.

k-NN graph is generated from a point cloud

We generate a k-NN graph from a point cloud.

Then, we discard the coordinates of nodes.
generate
edges
discard
coordinates
nodes have coordinates
for visualization
but they are random

Metric recovery asks to estimate the coodinates

The original coordinates are hidden now.

Metric recovery from unweighted k-NN graphs is a problem
of estimating the coordinates from the k-NN graph.
estimate

Only the existences of edges are observable

Unweighted means the edge lengths are neither available.

This is equivalent to the setting where only the 01-adjacency
matrix of the k-NN graph is available.
estimate

Given 01-adjacency, estimate the coordinates

Problem (Metric Recovery from Unweighted k-NN Graphs)
In: The 01-adjacency matrix of a k-NN graph
Out: The latent coordinates of the nodes

Very simple.
estimate

Why Is This Problem Challenging?

Standard node embedding methods fail

The type of this problem is node embedding.
I.e., In: graph, Out: node embeddings.

However, the following example tells standard embeddings
techniques fail.

Distance is opposite in the graph and latent space

The shortest-path distance between nodes A and B is 21.
The shortest-path distance between nodes A and C is 18.

Standard node embedding methods would embed node C
closer to A than node B to A, which is not consistent with
the ground truth latent coordinates.
10-NN graph
The coordinates are
supposed to be hidden,
but I show them for
illustration.

Critical assumption does not hold

Embedding nodes that are close in the input graph close
is the critical assumption in various embedding methods.

This assumption does NOT hold in our situation.
10-NN graph
The coordinates are
but I show them for
illustration.

Solution

Edge lengths are important

Why the previous example fails?

If the edge lengths were took into consideration,
the shortest path distance would be a consistent estimator of
the latent distance.

Step 1: Estimate the latent edge lengths.
10-NN graph
The coordinates are
but I show them for
illustration.

Densities are important

Observation: Edges are longer in sparse regions
and shorter in dense regions.

Step 2: Estimate the densities.

But how? We do not know the coordinates of the points...
10-NN graph
The coordinates are
but I show them for
illustration.

Density can be estimated from PageRank

Solution: A PageRank-like estimator solves it.
The stationary distribution of random walks (plus a simple
transformation) is a consistent estimator of the density.

The higher the rank is, the denser there is.

This can be computed solely from the unweighted graph.
10-NN graph
Stationary distribution
of simple random walks
≈ PageRank

Given 01-adjacency, estimate the coordinates

Problem definition (again)
In: The 01-adjacency matrix of a k-NN graph
Out: The latent coordinates of the nodes

Very simple.
estimate

Procedure to estimate the coordinates
1. Compute the stationary distribution of random walks.
2. Estimate the density around each node.
3. Estimate the edge lengths using the estimated densities.
4. Compute the shortest path distances using the estimated
edge lengths and compute the distance matrix.
5. Estimate the coordinates from the distance matrix
by, e.g., multidimentional scaling.

This is a consistent estimator [Hashimoto+ AISTATS 2015].
Tatsunori Hashimoto, Yi Sun, Tommi Jaakkola. Metric recovery from directed unweighted graphs. AISTATS 2015.
(up to rigid transform)

We can recover the coordinates consistently
The latent coordinates can be consistently estimated
solely from the unweighted k-NN graph.
Take Home Message

Towards Principled User-side Recommender Systems (CIKM 2022)
Ryoma Sato. Towards Principled User-side Recommender Systems. CIKM 2022.

Let’s consider item-to-item recommendations

We consider item-to-item recommendations.

Ex: “Products related to this item” panel in Amazon.com.

User-side recsys realizes user’s desiderata

Problem: We are unsatisfactory with the official recommender
system.

It provides monotone recommendations.
We need serendipity.

It provides recommendations biased towards specific
companies or countries.

User-side recommender systems [Sato 2022] enable users
to build their own recommender systems that satisfy their
desiderata even when the official one does not support them.
Ryoma Sato. Private Recommender Systems: How Can Users Build Their Own Fair Recommender Systems without
Log Data? SDM 2022.

We need powerful and principled user-side Recsys

[Sato 2022]’s user-side recommender system is realized in an
ad-hoc manner, and the performance is not so high.

We need a way to build user-side recommender systems in a
systematic manner and a more powerful one.
Hopefully one that is as strong as the official one.
Ryoma Sato. Private Recommender Systems: How Can Users Build Their Own Fair Recommender Systems without
Log Data? SDM 2022.

Official (traditional) recommender systems
Recsys
Algorithm
log data
catalog
auxiliary data
Ingredients
Recsys model
sourece item
Step 1. training
Step 2. inference
recommendations
Official (traditional) recsys

Users cannot see the data, algorithm, and model
Recsys
Algorithm
log data
catalog
auxiliary data
Ingredients
Recsys model
sourece item
recommendations
These parts are not
observable for users
(industrial secrets)

How can we build our Recsys without them?
Recsys
Algorithm
log data
catalog
auxiliary data
Ingredients
Recsys model
sourece item
recommendations
But they are crucial
information to build
new Recsys...

We assume the model is embedding-based
Recsys
Algorithm
log data
catalog
auxiliary data
Ingredients
Recsys model
sourece item
recommendations
(Slight) Assumption:
The model embeds items and
recommends near items.
This is a common strategy in Recsys.
We do not assume the way it embeds.
It can be matrix factorization,
neural networks, etc.

We can observe k-NN graph of the embeddings
Recsys
Algorithm
log data
catalog
auxiliary data
Ingredients
Recsys model
sourece item
recommendations
Observation:
These outputs have sufficient information
to construct the unweighted k-NN graph.
I.e., users can build the k-NN graph by
accessing each item page, and observing
what the neighboring items are.

We can estimate the embeddings!
Recsys
Algorithm
log data
catalog
auxiliary data
Ingredients
Recsys model
sourece item
recommendations
Solution:
Estimate the item embeddings of
the official Recsys.
They are considered to be secret,
but we can estimate them from
the weighted k-NN graph!
They contain much information!

We realize our desiderata with the embeddings

We can do many things with the estimated embeddings.

We can compute recommendations by ourselves and
with our own postprocessings.

If you want more serendipity,
recommend 1st, 2nd, 4th, 8th, ... and 32nd nearest items
or add noise to the embeddings.

If you want to decrease the bias to specific companies,
add negative biases to the score of these items so as to
suppress these companies.

Experiments validated the theory

In the experiments
I conducted simulations
and showed that the hidden
item embeddings can be
estimated accurately.
I built a fair Recsys for Twitter, which runs
in the real-world, on the user’s side.
Even though the official Recsys
is not fair w.r.t. gender, mine is, and
it is more efficient than the existing one.

Users can recover the item embeddings
Users can “reverse engineer” the official item
embeddings solely from the observable information.
Take Home Message

Graph Neural Networks can Recover the Hidden Features
Solely from the Graph Structure (ICML 2023)
Ryoma Sato. Graph Neural Networks can Recover the Hidden Features Solely from the Graph Structure. ICML 2023.

We call for the theory for GNNs

Graph Neural Networks (GNNs) take a graph with node
features as input and output node embeddings.

GNNs is a popular choice in various graph-related tasks.

GNNs are so popular that understanding GNNs by theory is
an important topic in its own right.
e.g., What is the hypothesis space of GNNs?
(GNNs do not have a universal approximation power.)
Why GNNs work well in so many tasks?

GNNs apply filters to node features

GNNs apply filters to the input node features and extract
useful features.

The input node features have long been considered
to be the key to success.
If the features have no useful signals, GNNs will not work.

Good node features are not always available

However, informative node features are not always available.

E.g., social network user information may be hidden for
privacy reasons.

Uninformative features degrade the performance

If we have no features at hand, we usually input
uninformative node features such as the degree features.

No matter how such features are filtered, only uninformative
embeddings are obtained.
“garbage in, garbage out.”
This is common sense.

Can GNNs work with uninformative node features?

Research question I want to answer in this project:
Do GNNs really not work when the input node features
are uninformative?

In practice, GNNs sometimes work just with degree features.
The reason is a mystery, which I want to elucidate.

We assume latent node features behind the graph

(Slight) Assumption:
The graph structure is formed by connecting nodes whose
latent node features z*
v
are close to each other.
 The latent node features z*
v
are not an observable
e.g., "true user preference vector"
Latent features that contain users’
preferences, workplace, residence, etc.
Those who have similar
preferences and residence
have connections.
We can only observe the way they are
connected, not the coordinates.

GNNs can recover the lantent feature

Main results:
GNNs can recover the latent node features z*
v
even when the
input node features are uninformative.
 z*
v
contains the preferences of users, which is useful for tasks.

GNNs create useful node features themselves

GNNs can create completely new and useful node
features by absorbing information from the graph structure,
even when the input node features are uninformative.

A new perspective that overturns the existing view of filtering
input node features.

GNNs can recover the coordinates with some tricks

How to prove it?
→ Metric recovery from k-NN graphs as you may expect.

But be careful when you apply it.
What GNNs can do (the hypothesis space of GNNs) is limited.

The metric recovery algorithm is compatible with GNNs.
Stationary distribution → GNNs can do random walks.
Shortest path → GNNs can simulate Bellman-Ford.
MDS → This is a bit tricky part. We send the matrix to
some nodes and solve it locally.

GNNs can recover the metric with slight additional errors.

Recovered features are empicirally useful

In the experiments,
We empirically confirmed
this phenomenon.
The recovered features are useful for various downstream tasks,
even when the input features xsyn
are uninformative.

GNNs can create useful features by themselves
GNNs can create useful node features by absorbing
information from the underlying graph.
Take Home Message

Conclusion

I introduced my favorite topic and its applications

Metric recovery from unweighted k-NN graphs is my
recent favorite technique.
I like this technique because
The scope of applications is broad, and
The results are simple but non-trivial.
The latent coordinates can be consistently estimated
solely from the unweighted k-NN graph.
Take Home Message

Metric Recovery from Unweighted k-NN Graphs

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