Monotone Bipartitions
This library enable manipulating and comparing implicitly defined
monotone bipartitions on the unit box. Namely, the user provides a
threshold oracle: oracle : [0, 1]^n -> bool with the constraint that
for any two points in the unit box, x, y in [0, 1]^n if x <= y
coordinate-wise, then oracle(x) <= oracle(y) , where False <= True. An example is given below:
The basis of the implemented algorithm to approximate the bipartition
using black box access to oracle was orignally given by Oded Maler
in Learning Monotone Partitions of Partially-Ordered
Domains.
Installation
If you just need to use monotone-partition, you can just run:
$ pip install monotone-bipartition
For developers, note that this project uses the poetry python package/dependency management tool. Please familarize yourself with it and then run:
$ poetry install
Usage
import monotone_bipartition as mbp
partition1 = mbp.from_threshold(
func=lambda x: x[0] >= 0.5,
dim=2,
) # type: mbp.BiPartition
assert partition1.dim == 2
# Approximate the boundary using a collection of rectangles.
recs = partition1.approx(tol=1e-3) # List of rectangles.
## Rectangles are defined as the product of intervals.
## I.e, each interval is the projection of the rectangle
## on a given axis of the unit box.
print(recs[0].intervals) # (Interval(bot=0.49999237060546875, top=0.5), Interval(bot=0.0, top=1)
# Support labeling point using boundary.
# Useful for testing equiv to `oracle` or
# if calling `oracle` is very expensive.
assert partition1.label((0.8, 0))
assert not partition1.label((0.3, 0.3))Comparing partitions
It is often useful to compare boundaries. (See https://github.com/mvcisback/LogicalLens for a usecase).
d11 = partition1.dist(partition1, tol=1e-1) # Returns an Interval
assert 0 in d12
assert d11.radius <= tol
print(d11.center) # 0.029
partition2 = mbp.from_threshold(
func=lambda x: x[1] >= 0.6,
dim=2,
) # type: mbp.BiPartition
d12 = partition1.dist(partition2, tol=1e-1) # Returns an Interval
assert 0.6 in d12
assert d12.radius <= tol
print(d12.center) # 0.5726
# TODO: implement partial ordering. Check if lower sets are subsets of each other.
partition3 = mbp.from_threshold(func=lambda x: x[1] >= 0.7, dim=2)
assert partition3 >= partition2
assert not (partition1 >= partition3) # Incomparable since they intersect.Find particular points on the boundary
Sometimes, it is useful to find particular points on the threshold boundary. For example, the following two works leverage such "projections" to learn classifiers for the data that induces the partitions. (Again, see https://github.com/mvcisback/LogicalLens).
# Find the point that intersects the line running
# between the origin and (0.2, 0.2).
rec = partition1.project((0.2, 0.2)) # Approximation of intersection point.
assert rec.shortest_edge < 1e-3
x = rec.center # Can use the center of the rectangle as the projected point.
## This value is approx (0.5, 0.5)
assert abs(max(x[0] - 0.5, x[1] - 0.5)) < 1e-3
# Find the point that lexicographically maximizes the 0-axis
# and then miniminizes the 1-axis.
order = [(1, True), (0, False)] # List of axis index and should_max tuples.
rec = partition1.project(order, lexicographic=True)
assert rec.shortest_edge < 1e-3
x = rec.center # Can use the center of the rectangle as the projected point.
## This value is approx (0.5, 1)
assert abs(max(x[0] - 0.5, x[1] - 1)) < 1e-3Intersection Strategy
By default, the find_threshold function uses a binary search along
the diagonal (with a fixed relative tolerance) to find asub-division
point. This can be overridden using the find_intersect argument. For
example, below we change the binary search tolerance to 1e-2.
from typing import Tuple
from monotone_bipartition.rectangles import Rec
from monotone_bipartition.search import binsearch, SearchResultType
def find_intersect(domain: Rec, oracle) -> Tuple[SearchResultType, Rec]:
"""Returns a rectangle within the domain containing the boundary.
This rectangle will be used to update the boundary approximation
during subdivision.
"""
# eps is the relative tolerance.
result_type, rec = binsearch(domain, oracle, eps=1e-2)
return result_type, rec
partition1 = mbp.from_threshold(
func=lambda x: x[0] >= 0.5,
dim=2,
find_intersect=find_intersect,
) # type: mbp.BiPartition
