Algebraic Property Testing:
A Unified Perspective
Arnab Bhattacharyya
Indian Institute of Science
Property Testing
Distinguish between
and
and
Property P
𝜖-far from property P
Testing and Learning

Proper learning (with membership queries) is as hard
as testing, for any property

For many natural properties, testing is much easier
than learning

Learning always requires time at least as large as size
of hypothesis but testing can be in constant time!
A brief history

Initially appeared as a tool for program checking [BlumLuby-Rubinfeld ‘90]

[Babai-Fortnow-Lund ’90, Rubinfeld-Sudan ’96]: application
to PCPs and low-degree testing

[Goldreich-Goldwasser-Ron ‘98] considered property testing
as full-fledged computational problem in its own right

Now, many other variants and connections known (e.g.,
implicit learning, active testing, coding theory,
inapproximability, …)
Testable Properties
Property P is testable if there exist functions 𝑞 and 𝛿 and a
randomized algorithm A such that, given an input object
and a parameter 𝜖:
•
A makes 𝑞(𝜖) queries into input object
•
A rejects with prob. ≥ 𝛿(𝜖) if input is 𝜖-far from P and
with prob. 0 if 𝜖 = 0
𝛿
Positive at
𝜖>0
Zero at 𝜖 = 0
ϵ
Testing all-orangeness
P = {all-orange rectangle}
• 𝑞=1
• Query hits non-orange region with
probability ≥ 𝜖
Testing linearity
P = {linear functions 𝐿: {𝟎, 𝟏}𝑛 → {𝟎, 𝟏} }
000…
0
000…
1
001…
1
001…
1
010…
1
010…
0
100…
0
100…
1
101…
1
111…
0
111…
0
Functions of the form
101…
1
𝑳 𝒙 , … , 𝒙𝒏 = 𝒙𝒊𝟏 + 𝒙𝒊𝟐 + 𝒙110…
𝒊𝟑 + ⋯0+ 𝒙𝒊𝒌
110… 𝟏 1
over characteristic011…
2
011…
0
1
• 𝑞=3
• 𝑓 𝑥 + 𝑓 𝑦 ≠ 𝑓 𝑥 + 𝑦 with probability Ω(𝜖). [BlumLuby-Rubinfeld]
Testing low-degree
P = {polynomials 𝑃: 𝑭𝒑𝑛 → 𝑭𝒑 of degree ≤ 𝑑} }
Functions of the form
𝑷 𝒙𝟏 , … , 𝒙𝒏 =
𝒊
𝒊
𝒊
𝒄𝒊𝟏 ⋯𝒊𝒏 ⋅ 𝒙𝟏𝟏 𝒙𝟐𝟐 ⋯ 𝒙𝒏𝒏
Check that function restricted
to a random O(1)-dim
𝒊𝟏 +⋯+𝒊𝒏 ≤𝒅
subspace is poly of degree ≤ 𝑑
over characteristic p
Test detects a violation with probability Ω(𝜖) [KaufmanRon, Haramaty-Shpilka-Sudan]
Affine-invariant properties
Subject of this talk: “algebraic properties” of
multivariate functions over finite fields.
Definition (affine-invariant property):
If a function f on domain 𝑭𝒏𝒑 satisfies an
affine-invariant property P, then 𝑓 ∘ 𝐴 must
also satisfy P for every affine map 𝐴: 𝑭𝒏𝒑 →
𝑭𝒏𝒑
[Kaufman-Sudan]
Why?
Introspective:
 A principled understanding of testability (~
VC dimension theory for PAC learning)
 What kinds of tools are needed to prove
testability? Limitations?
Extrospective:
 Search for locally testable codes (applications
to inapproximability)
Testing Fourier Sparsity
P = {functions 𝑃: 0,1 𝑛 → {0,1} have at
most k non-zero Fourier coefficients}
[Gopalan-O’Donnell-Servedio-Shpilka-Wimmer]
showed P is testable (though only with 𝛿 0 > 0).
Testing Decompositions

P = {functions 𝑃: 𝑭𝒑𝑛 → 𝑭𝒑 that are a product of two
quadratics}

P = {functions 𝑃: 𝑭𝒑𝑛 → 𝑭𝒑 that are the square of a
quartic}

P = {polynomials 𝑃: 𝑭𝒑𝑛 → 𝑭𝒑 of the form 𝑎𝑏 + 𝑐𝑑
where 𝑎, 𝑏, 𝑐, 𝑑 are all cubics}
Testability of all such properties open previously!
Main Result
An exact characterization of testability for affineinvariant properties:
Theorem: An affine-invariant property is
testable* if and only if it is locally
characterized
[Joint work with Fischer, H&P Hatami, Lovett ‘13]
Locally Characterized
Definition (locally characterized):
A property P is locally characterized if there
always exists a constant-sized witness to
non-membership in P.
Examples:
• Linearity: If 𝑓 isn’t linear, then there exist 𝑥, 𝑦 such that
𝑓 𝑥 + 𝑓 𝑦 ≠ 𝑓(𝑥 + 𝑦).
• Low degree: If deg 𝑓 > 𝑑, then there exists a point at
which the (𝑑 + 1)-th order derivative is nonzero.
Necessity of locality
Theorem: An affine-invariant property is
testable if and only if it is locally
characterized.
Proof of : The set of queries by the tester that
make it reject is itself a 𝑞-sized witness to nonmembership in the property.
Proof of
: Rest of this talk…
Degree-structural properties

Are properties like “Fourier sparsity”, “product
of two quadratics”, “square of a quartic”, “sum
of two products of quadratics”, “splitting into d
linear forms” locally characterized?
Theorem: Any property described as the
property of decomposing into a known
structure of low-degree polynomials is
locally characterized.
Subsequent work

Degree-structural properties are not only
testable but reconstructible!
◦ Given degree-𝑑 poly 𝑃 = 𝑄1 𝑄2 + 𝑄3 𝑄4 where
𝑄1 , … , 𝑄4 are degree-(𝑑/2) polys, we can
reconstruct 𝑄1 , … , 𝑄4 by making 𝑂(𝑛𝑑 ) queries to
𝑃. [B. ‘14]

Characterization of two-sided testability
[Yoshida ‘14]
Some drawbacks…
Unfolded proof uses
non-constructive
results. We get no
bound on the locality
and query
complexity for even
simple properties
like “product of two
quadratics”!
A Running Example: RBG
𝑥+𝑦
𝑥+𝑦+𝑧
Witness to non-membership:
𝑥
𝑥+𝑧
RBG square
A function 𝑓: 𝑭𝑛 → {𝑅𝑒𝑑, 𝐵𝑙𝑢𝑒, 𝐺𝑟𝑒𝑒𝑛} satisfies the RBG
property if there are no 𝑥, 𝑦, 𝑧 ∈ 𝑭𝑛 such that 𝑓 𝑥 =
𝑓 𝑥 + 𝑦 = 𝑅𝑒𝑑, 𝑓 𝑥 + 𝑧 = 𝐵𝑙𝑢𝑒, and 𝑓 𝑥 + 𝑦 + 𝑧 =
𝐺𝑟𝑒𝑒𝑛.
The RBG claim
Claim: The RBG property is testable with 4
queries.
Suffices to show that if 𝑓: 𝑭𝑛 →
{𝑅𝑒𝑑, 𝐵𝑙𝑢𝑒, 𝐺𝑟𝑒𝑒𝑛} is far from
RBG, then a random tuple
(𝑥, 𝑥 + 𝑦, 𝑥 + 𝑧, 𝑥 + 𝑦 + 𝑧) is an
RBG square with constant
probability.
𝑥+𝑦
𝑥
𝑥+𝑦+𝑧
𝑥+𝑧
Dreams of a proof
(Image © Kozmic Konstructions)
The dreamiest situation
Suppose non-RBG function 𝑓 has the form:
𝑓(𝑥) = Γ 𝑥1 , 𝑥2 , … , 𝑥𝑐
(0,4)
• 𝑝𝑐 cells of equal size
(0,3)
(0,2)
(0,1)
(0,0)
• Must exist at least one
RBG square
• Probability of selecting
an RBG square is 𝑝−3𝑐 .
A slight nudge
Same analysis works if non-RBG function 𝑓
has the form:
𝑓(𝑥) = Γ ℓ1 𝑥 , ℓ2 𝑥 , … , ℓ𝑐 (𝑥)
where ℓ1 , ℓ2 , … , ℓ𝑐 are linearly independent
linear forms.
Also works if “linearly
independent” replaced
by “random”
More rocking of the bed
Suppose non-RBG function 𝑓 has the form:
𝑓(𝑥) = Γ 𝑄1 𝑥 , 𝑄2 𝑥 , … , 𝑄𝑐 (𝑥)
where 𝑄1 , … , 𝑄𝑐 are random bounded degree nonlinear polynomials.
Partitioning by random polys
(0,4)
(0,3)
Joint distribution of
𝑄1 , … , 𝑄𝑐
close to uniform
distribution
(0,2)
(0,1)
(0,0)
𝑝𝑐 cells of roughly
equal size
Partitioning by random polys
(0,4)
(0,3)
(0,2)
Joint distribution of
(𝑄1 (𝑥), . . , 𝑄𝑐 (𝑥), 𝑄1 (𝑥 +
𝑦), . . . , 𝑄𝑐 (𝑥 + 𝑦), 𝑄1 (𝑥 +
𝑧), . . . , 𝑄𝑐 (𝑥 + 𝑧), 𝑄1 (𝑥 +
𝑦 + 𝑧), . . . , 𝑄𝑐 (𝑥 + 𝑦 +
𝑧))close to uniform
(0,1)
(0,0)
Probability of selecting an
RBG square is ≈ 𝑝−3𝑐 .
Returning to intermediate dream
Suppose non-RBG function 𝑓 has the form:
𝑓(𝑥) = Γ 𝑄1 𝑥 , 𝑄2 𝑥 , … , 𝑄𝑐 (𝑥)
where 𝑄1 , … , 𝑄𝑐 are arbitrary low degree polynomials.
Instead of insisting 𝑄1 , … , 𝑄𝑐 be truly random, can
we weaken the requirement?
Returning to intermediate dream
Suppose non-RBG function 𝑓 has the form:
𝑓(𝑥) = Γ 𝑄1 𝑥 , 𝑄2 𝑥 , … , 𝑄𝑐 (𝑥)
where 𝑄1 , … , 𝑄𝑐 are arbitrary low degree polynomials.
How to ensure (𝑄1 , … , 𝑄𝑐 ) distributed close to
uniform? How to even ensure 𝑄1 distributed close
to uniform?
High rank polynomials
Theorem [Green-Tao, Kaufmann-Lovett]: A
polynomial 𝑄: 𝐹 𝑛 → 𝐹 is distributed close to
uniform if Q is of high rank.
Definition: A polynomial 𝑄: 𝐹 𝑛 → 𝐹 is of rank > 𝑟 if
there are no polynomials 𝑃1 , … , 𝑃𝑟 of degrees less than
deg(Q) such that
𝑄 = Λ 𝑃1 , … , 𝑃𝑟
for some function Λ.
High rank polynomial collection
Theorem [Green-Tao, Kaufmann-Lovett]:
Polynomials 𝑄1 , … , 𝑄𝑐 : 𝐹 𝑛 → 𝐹 jointly distributed
close to uniform if every nontrivial linear
combination of 𝑄1 , … , 𝑄𝑐 is high rank.
Polynomial Regularity Lemma
A straightforward inductive argument shows that,
given
𝑓(𝑥) = Γ 𝑄1 𝑥 , 𝑄2 𝑥 , … , 𝑄𝑐 (𝑥)
we can find a high rank collection of polynomials
𝑃1 , … , 𝑃𝑐 ′ such that:
𝑓(𝑥) = Γ′ 𝑃1 𝑥 , 𝑃2 𝑥 , … , 𝑃𝑐 ′ (𝑥)
𝑃1 , … , 𝑃𝑐 ′ form cells of roughly equal size
Equidistribution of squares?
Recall we also wanted equidistribution of squares. Is
high rank sufficient for this purpose?
Wanted Lemma: If 𝑃1 , … , 𝑃𝑐 ′ of sufficiently high
rank, then:
𝑃1 𝑥 , . . , 𝑃𝑐 ′ 𝑥 ,
𝑃1 𝑥 + 𝑦 , . . . , 𝑃𝑐 ′ 𝑥 + 𝑦 ,
𝑃1 𝑥 + 𝑧 , . . . , 𝑃𝑐 ′ 𝑥 + 𝑧 ,
𝑃1 𝑥 + 𝑦 + 𝑧 , . . . , 𝑃𝑐 ′ 𝑥 + 𝑦 + 𝑧
distributed close to uniform.
A bit of a nightmare…
Wanted lemma not necessarily true!
Suppose 𝑃1 is of degree 1. Then:
𝑃1 𝑥 − 𝑃1 𝑥 + 𝑦 − 𝑃1 𝑥 + 𝑧 + 𝑃1 𝑥 + 𝑦 + 𝑧 = 0
identically.
Lesson: some correlations may be present because
of degree, no matter how large the rank is
Corrected equidistribution
Lemma: Given polynomials 𝑃1 , … , 𝑃𝑐 of high rank,
either there’s some linear identity among
𝑃𝑖 𝑥 , 𝑃𝑖 𝑥 + 𝑦 , 𝑃𝑖 𝑥 + 𝑧 , 𝑃𝑖 (𝑥 + 𝑦 + 𝑧) for some i,
or the tuple
𝑃1 𝑥 , . . , 𝑃𝑐 𝑥 ,
𝑃1 𝑥 + 𝑦 , . . . , 𝑃𝑐 𝑥 + 𝑦 ,
𝑃1 𝑥 + 𝑧 , . . . , 𝑃𝑐 𝑥 + 𝑧 ,
𝑃1 𝑥 + 𝑦 + 𝑧 , . . . , 𝑃𝑐 𝑥 + 𝑦 + 𝑧
is close to uniformly distributed.
Proof uses “Gowers
inverse theorem”
[Tao-Ziegler]
Applying corrected equidistribution
𝟐
𝟎
2
2
𝟑
4
𝟐
𝟎
𝟐
𝟒
𝟑
Suppose 𝑃1 satisfies: 𝑃1 𝑥 −
𝑃1 𝑥 + 𝑦 − 𝑃1 𝑥 + 𝑧 + 𝑃1 (𝑥 +
Almost awake to reality…
Now, let’s remove all restrictions on RBG-far
function 𝑓: 𝑭𝑛 → {𝑅𝑒𝑑, 𝐵𝑙𝑢𝑒, 𝐺𝑟𝑒𝑒𝑛}.
Can we “approximate” f by some function g of
the form:
𝑔(𝑥) = Γ 𝑄1 𝑥 , 𝑄2 𝑥 , … , 𝑄𝑐 (𝑥)
where 𝑄1 , … , 𝑄𝑐 are low degree polynomials?
Regularity Lemma
Given any function 𝑓: 𝑭𝑛 → {0,1} and integer 𝑑, there exist
constantly many polynomials 𝑄1 , … , 𝑄𝑐 : 𝑭𝑛 → 𝑭 of degrees at
most 𝑑, a function Γ: 𝑭𝑐 → [0,1] and a function ℎ: 𝑭𝑛 → [−1,1]
such that:
• 𝑓 𝑥 = Γ 𝑄1 𝑥 , … , 𝑄𝑐 𝑥
+ℎ 𝑥
• Collection 𝑄1 , … , 𝑄𝑐 has high rank
Uses “Gowers
inverse theorem”
[Tao-Ziegler]
• Γ(𝑎1 , … , 𝑎𝑐 ) is the average value of 𝑓 on the cell indexed
(𝑎1 , … , 𝑎𝑐 )
• (𝑑 + 1)-th order Gowers norm of h is small
Victory in dreamland!

Letting 𝑓 𝑐 : 𝐹 𝑛 → {0,1} be indicator function of color c.
Then, rejection probability is:
𝐸𝑥,𝑦,𝑧 [𝑓 𝑟𝑒𝑑 𝑥 𝑓 𝑟𝑒𝑑 𝑥 + 𝑦 𝑓 𝑏𝑙𝑢𝑒 𝑥 + 𝑧 𝑓 𝑔𝑟𝑒𝑒𝑛 𝑥 + 𝑦 + 𝑧 ]

We apply the regularity lemma. The term with low
Gowers norm contributes negligibly to expectation.

A combinatorial argument finishes the claim.
Waking up to hard reality

“Inverse conjecture for the Gowers norm is false” [LovettMeshulam-Samorodnitsky]

Turns out that in the conjecture, one needs to modify the
definition of polynomials to include functions that are not 𝑭valued.

Need to reprove regularity lemma and equidistribution
claims for such “non-classical polynomials” (technical core of
our work)

Also, several lies encountered in this dream sequence
addressed
Non-classical polynomials
Definition (non-classical polynomial):
A function 𝑓: 𝑭𝒏𝒑 → ℝ/ℤ is of degree ≤ 𝑑 if
Δℎ1 Δℎ2 ⋯ Δℎ𝑑+1 𝑓 ≡ 0
for any ℎ1 , ℎ2 , … , ℎ𝑑+1 ∈ 𝑭𝒏𝒑 , where:
Δℎ 𝑓 𝑥 = 𝑓 𝑥 + ℎ − 𝑓(𝑥)
• If range of 𝑓 is
1 2
𝑝−1
0, , , … ,
𝑝 𝑝
𝑝
same as classical
, non-classical
Non-classical polynomials
Definition (non-classical polynomial):
A function 𝑓: 𝑭𝒏𝒑 → ℝ/ℤ is of degree ≤ 𝑑 if
Δℎ1 Δℎ2 ⋯ Δℎ𝑑+1 𝑓 ≡ 0
for any ℎ1 , ℎ2 , … , ℎ𝑑+1 ∈ 𝑭𝒏𝒑 , where:
Δℎ 𝑓 𝑥 = 𝑓 𝑥 + ℎ − 𝑓(𝑥)
1 1 3
4 2 4
• Function 𝑓: 𝐹2 → 0, , ,
defined as
1
𝑓 0 = 0, 𝑓 1 =
4
is non-classical poly of degree 2.
Gowers norm and inverse theorem

For functions 𝑓: 𝐹𝑝𝑛 → ℂ, the Gowers norm of order d measures
the expected multiplicative derivative of 𝑓 at a random point
in 𝑑 random directions.

Fact: If Gowers norm of f of order d is 1 and f takes values
inside the unit disk, then f is the exponential of a non-classical
polynomial of degree ≤ 𝑑 − 1.

Gowers Inverse Theorem: If Gowers norm of f of order d is
𝛿 > 0 and f takes values inside the unit disk, then f is
correlated with a non-classical polynomial of degree ≤ 𝑑 − 1.
Open Questions

Proof for degree-structural property also uses the same
core technical ideas, so no good locality bounds even for
“simple” properties like product of two quadratics.

Give non-trivial lower bound for testing any natural
affine-invariant property. No superpolynomial lower
bounds known in 1/𝜖.

Find other uses of equidistribution of high rank
polynomials.
THANK YOU!