Bad Science Matrix Problem

• Bad Science Matrix Problem is defined as maximizing the average largest coordinate of Ax over all sign-vectors x with each row of A constrained to unit ℓ2-norm.
• The analysis employs probabilistic bounds, concentration, and anti-concentration inequalities to establish the asymptotic behavior, notably the √(2 log n) growth rate.
• Explicit constructions using inductive techniques and almost-Hadamard matrices reveal structured, low-dimensional extremizers that nearly attain optimal performance.

The Bad Science Matrix Problem is the extremal question of maximizing the average largest coordinate of $Ax$ over all sign-vectors $x\in\{\pm1\}^n$, subject to the constraint that every row of $A\in\mathbb{R}^{n\times n}$ has unit $\ell^2$-norm. In Steinerberger’s formulation, if the rows are $a_1,\dots,a_n$, then

$$F(A)=\frac1{2^n}\sum_{x\in\{-1,1\}^n}\|Ax\|_\infty,$$

and later papers write the same quantity as

$$\beta(A)=\frac1{2^n}\sum_{x\in\{\pm1\}^n}\|Ax\|_\infty =\mathbb{E}_{x\sim\{\pm1\}^n}\bigl[\max_{1\le i\le n}|\langle a_i,x\rangle|\bigr].$$

The problem asks how large this average can be under the row-normalization constraint $\|a_i\|_2=1$ for all $i$ (Steinerberger, 2024).

1. Formal optimization problem and statistical analogy

The admissible class is

$$S_n=\{A\in\mathbb{R}^{n\times n}:\text{ each row }a_i\text{ satisfies }\|a_i\|_2=1\},$$

and the extremal value is

$x\in\{\pm1\}^n$0

Geometrically, $x\in\{\pm1\}^n$1 maps the discrete cube $x\in\{\pm1\}^n$2 to $x\in\{\pm1\}^n$3 points in $x\in\{\pm1\}^n$4, and $x\in\{\pm1\}^n$5 is the average of the largest-coordinate magnitudes of those image points (Albors et al., 2024). Steinerberger also describes these matrices as corresponding to affine transformations of the discrete unit cube to points with, on average, at least one large coordinate (Steinerberger, 2024).

The name of the problem comes from a statistical analogy. One interprets $x\in\{\pm1\}^n$6 as the outcomes of $x\in\{\pm1\}^n$7 independent fair coin tosses. A single test is given by a unit vector $x\in\{\pm1\}^n$8, with test statistic $x\in\{\pm1\}^n$9. By Hoeffding’s inequality, $A\in\mathbb{R}^{n\times n}$0 is sub-Gaussian, so large values are unlikely under the null. A “dishonest” scientist instead fixes $A\in\mathbb{R}^{n\times n}$1 tests $A\in\mathbb{R}^{n\times n}$2, observes $A\in\mathbb{R}^{n\times n}$3, and reports $A\in\mathbb{R}^{n\times n}$4. The quantity $A\in\mathbb{R}^{n\times n}$5 or $A\in\mathbb{R}^{n\times n}$6 is exactly the expected size of this maximal test statistic, and therefore measures the extent to which the maximum over many fair tests typically produces an atypical large value (Steinerberger, 2024).

A recurring misconception is to treat the problem as only a metaphor about $A\in\mathbb{R}^{n\times n}$7-hacking. The papers formulate it as a precise extremal problem in discrete probability, convex geometry, and high-dimensional analysis; the “bad scientist” language is an interpretation of the optimization objective, not a substitute for it (Steinerberger, 2024).

2. Asymptotic growth and proof mechanisms

The central asymptotic theorem is that

$A\in\mathbb{R}^{n\times n}$8

Equivalently,

$A\in\mathbb{R}^{n\times n}$9

as (n\to\infty) (Steinerberger, 2024, Sinha, 11 Sep 2025).

The upper bound is obtained by concentration. For any fixed row $\ell^2$0 with $\ell^2$1, the random variable $\ell^2$2 satisfies

$\ell^2$3

Taking $\ell^2$4 gives $\ell^2$5. A union bound over the $\ell^2$6 rows yields

$\ell^2$7

and splitting the expectation into the typical event $\ell^2$8 and the rare tail gives the desired asymptotic upper bound (Steinerberger, 2024).

The lower bound is established probabilistically. If $\ell^2$9 has iid entries $a_1,\dots,a_n$0, then each row has unit $a_1,\dots,a_n$1-norm. For fixed $a_1,\dots,a_n$2, each coordinate $a_1,\dots,a_n$3 is a rescaled Binomial random variable. Using an anti-concentration result identified as Tusnády’s lemma, Steinerberger shows that for any small $a_1,\dots,a_n$4,

$a_1,\dots,a_n$5

for some $a_1,\dots,a_n$6. Independence of rows then implies that

$a_1,\dots,a_n$7

and a double-counting or Fubini argument over all $a_1,\dots,a_n$8 sign-vectors produces a single matrix $a_1,\dots,a_n$9 for which all but $$F(A)=\frac1{2^n}\sum_{x\in\{-1,1\}^n}\|Ax\|_\infty,$$0 of the $$F(A)=\frac1{2^n}\sum_{x\in\{-1,1\}^n}\|Ax\|_\infty,$$1 satisfy the same lower bound on $$F(A)=\frac1{2^n}\sum_{x\in\{-1,1\}^n}\|Ax\|_\infty,$$2 (Steinerberger, 2024).

The scale $$F(A)=\frac1{2^n}\sum_{x\in\{-1,1\}^n}\|Ax\|_\infty,$$3 is the classical scale for the maximum of $$F(A)=\frac1{2^n}\sum_{x\in\{-1,1\}^n}\|Ax\|_\infty,$$4 approximately independent Gaussians. Steinerberger explicitly notes that $$F(A)=\frac1{2^n}\sum_{x\in\{-1,1\}^n}\|Ax\|_\infty,$$5, which explains why the same growth appears here (Steinerberger, 2024).

3. Explicit constructions and low-dimensional extremizers

Beyond existential lower bounds, explicit constructions are known. Albors, Bhatti, Ganjoo, Guo, Kunisky, Mukherjee, Stepin, and Zeng construct explicit $$F(A)=\frac1{2^n}\sum_{x\in\{-1,1\}^n}\|Ax\|_\infty,$$6 matrices $$F(A)=\frac1{2^n}\sum_{x\in\{-1,1\}^n}\|Ax\|_\infty,$$7 with

$$F(A)=\frac1{2^n}\sum_{x\in\{-1,1\}^n}\|Ax\|_\infty,$$8

and describe this as only $$F(A)=\frac1{2^n}\sum_{x\in\{-1,1\}^n}\|Ax\|_\infty,$$9 smaller than the asymptotic rate (Albors et al., 2024). Their construction is inductive. Starting from

$$\beta(A)=\frac1{2^n}\sum_{x\in\{\pm1\}^n}\|Ax\|_\infty =\mathbb{E}_{x\sim\{\pm1\}^n}\bigl[\max_{1\le i\le n}|\langle a_i,x\rangle|\bigr].$$0

if $$\beta(A)=\frac1{2^n}\sum_{x\in\{\pm1\}^n}\|Ax\|_\infty =\mathbb{E}_{x\sim\{\pm1\}^n}\bigl[\max_{1\le i\le n}|\langle a_i,x\rangle|\bigr].$$1 has $$\beta(A)=\frac1{2^n}\sum_{x\in\{\pm1\}^n}\|Ax\|_\infty =\mathbb{E}_{x\sim\{\pm1\}^n}\bigl[\max_{1\le i\le n}|\langle a_i,x\rangle|\bigr].$$2, they define

$$\beta(A)=\frac1{2^n}\sum_{x\in\{\pm1\}^n}\|Ax\|_\infty =\mathbb{E}_{x\sim\{\pm1\}^n}\bigl[\max_{1\le i\le n}|\langle a_i,x\rangle|\bigr].$$3

and prove that $$\beta(A)=\frac1{2^n}\sum_{x\in\{\pm1\}^n}\|Ax\|_\infty =\mathbb{E}_{x\sim\{\pm1\}^n}\bigl[\max_{1\le i\le n}|\langle a_i,x\rangle|\bigr].$$4. Iterating yields explicit matrices $$\beta(A)=\frac1{2^n}\sum_{x\in\{\pm1\}^n}\|Ax\|_\infty =\mathbb{E}_{x\sim\{\pm1\}^n}\bigl[\max_{1\le i\le n}|\langle a_i,x\rangle|\bigr].$$5 with

$$\beta(A)=\frac1{2^n}\sum_{x\in\{\pm1\}^n}\|Ax\|_\infty =\mathbb{E}_{x\sim\{\pm1\}^n}\bigl[\max_{1\le i\le n}|\langle a_i,x\rangle|\bigr].$$6

hence $$\beta(A)=\frac1{2^n}\sum_{x\in\{\pm1\}^n}\|Ax\|_\infty =\mathbb{E}_{x\sim\{\pm1\}^n}\bigl[\max_{1\le i\le n}|\langle a_i,x\rangle|\bigr].$$7 (Albors et al., 2024).

Exact optima are known for $$\beta(A)=\frac1{2^n}\sum_{x\in\{\pm1\}^n}\|Ax\|_\infty =\mathbb{E}_{x\sim\{\pm1\}^n}\bigl[\max_{1\le i\le n}|\langle a_i,x\rangle|\bigr].$$8, while for $$\beta(A)=\frac1{2^n}\sum_{x\in\{\pm1\}^n}\|Ax\|_\infty =\mathbb{E}_{x\sim\{\pm1\}^n}\bigl[\max_{1\le i\le n}|\langle a_i,x\rangle|\bigr].$$9 the literature gives candidate maximizers and best known values. The following values all appear in the cited papers (Steinerberger, 2024, Albors et al., 2024).

$\|a_i\|_2=1$0	Value	Status
$\|a_i\|_2=1$1	$\|a_i\|_2=1$2	exact
$\|a_i\|_2=1$3	$\|a_i\|_2=1$4	exact
$\|a_i\|_2=1$5	$\|a_i\|_2=1$6	exact
$\|a_i\|_2=1$7	$\|a_i\|_2=1$8	exact
$\|a_i\|_2=1$9	$i$0	candidate / best known lower bound
$i$1	$i$2	candidate / best known lower bound
$i$3	$i$4	candidate / best known lower bound
$i$5	$i$6	candidate / best known lower bound

Several low-dimensional matrices have especially rigid forms. For $i$7, the maximizer is unique up to rotation by $i$8, namely

$i$9

and it maps the square corners to $$S_n=\{A\in\mathbb{R}^{n\times n}:\text{ each row }a_i\text{ satisfies }\|a_i\|_2=1\},$$0 and $$S_n=\{A\in\mathbb{R}^{n\times n}:\text{ each row }a_i\text{ satisfies }\|a_i\|_2=1\},$$1 (Steinerberger, 2024). For $$S_n=\{A\in\mathbb{R}^{n\times n}:\text{ each row }a_i\text{ satisfies }\|a_i\|_2=1\},$$2, one optimal matrix is

$$S_n=\{A\in\mathbb{R}^{n\times n}:\text{ each row }a_i\text{ satisfies }\|a_i\|_2=1\},$$3

for which half of the cube vertices are sent to max-coordinate $$S_n=\{A\in\mathbb{R}^{n\times n}:\text{ each row }a_i\text{ satisfies }\|a_i\|_2=1\},$$4 and half to $$S_n=\{A\in\mathbb{R}^{n\times n}:\text{ each row }a_i\text{ satisfies }\|a_i\|_2=1\},$$5 (Steinerberger, 2024, Albors et al., 2024). For $$S_n=\{A\in\mathbb{R}^{n\times n}:\text{ each row }a_i\text{ satisfies }\|a_i\|_2=1\},$$6, exact optima include matrices that ignore one coordinate and satisfy $$S_n=\{A\in\mathbb{R}^{n\times n}:\text{ each row }a_i\text{ satisfies }\|a_i\|_2=1\},$$7; Steinerberger also records an orthogonal maximizer with $$S_n=\{A\in\mathbb{R}^{n\times n}:\text{ each row }a_i\text{ satisfies }\|a_i\|_2=1\},$$8 that sends every sign-vector to an image with $$S_n=\{A\in\mathbb{R}^{n\times n}:\text{ each row }a_i\text{ satisfies }\|a_i\|_2=1\},$$9 (Steinerberger, 2024, Albors et al., 2024). For $x\in\{\pm1\}^n$00, the transpose of the discrete Haar-wavelet transform of size $x\in\{\pm1\}^n$01 sends each of the $x\in\{\pm1\}^n$02 sign-vectors to one of the $x\in\{\pm1\}^n$03 basis directions at coordinate $x\in\{\pm1\}^n$04, giving $x\in\{\pm1\}^n$05 (Steinerberger, 2024).

The small-$x\in\{\pm1\}^n$06 examples are notable because they are not generic random sign matrices. Steinerberger emphasizes that the extremizers up to $x\in\{\pm1\}^n$07 appear highly structured, often combinatorial or wavelet-like, and in dimensions $x\in\{\pm1\}^n$08 the best found matrices can have entire columns of zeros while still outperforming random matrices numerically up to $x\in\{\pm1\}^n$09 (Steinerberger, 2024).

4. Algebraic and combinatorial structure of extremizers

A structural theorem in (Albors et al., 2024) gives an explicit description of the rows of an extremal matrix. If $x\in\{\pm1\}^n$10 maximizes $x\in\{\pm1\}^n$11, and if for each row $x\in\{\pm1\}^n$12 one defines

$x\in\{\pm1\}^n$13

then

$x\in\{\pm1\}^n$14

The proof rewrites $x\in\{\pm1\}^n$15 by partitioning the cube into the $x\in\{\pm1\}^n$16, applies Cauchy–Schwarz on each block, and shows that any strict inequality can be improved by rescaling the corresponding row (Albors et al., 2024).

A direct corollary is algebraicity. Since each $x\in\{\pm1\}^n$17 has integer coordinates, every entry of $x\in\{\pm1\}^n$18 is of the form

$x\in\{\pm1\}^n$19

so extremal entries are square-roots of rationals (Albors et al., 2024). The abstract of the same paper summarizes this by stating that every entry of any optimal matrix is a square root of a rational number (Albors et al., 2024).

This characterization turns the continuous optimization problem into a finite combinatorial problem once the winning sets $x\in\{\pm1\}^n$20 are specified. That reduction underlies the finite-state computer-assisted search proving exact optimality for $x\in\{\pm1\}^n$21 (Albors et al., 2024). A plausible implication is that the qualitative rigidity observed in small dimensions is not accidental: the extremal rows are constrained to be normalized centroids of discrete subsets of the cube.

5. Geometric, Fourier-analytic, and Gaussian reformulations

Subsequent work gives a more geometric description of near-extremizers. For each row index $x\in\{\pm1\}^n$22, define the full winning cell

$x\in\{\pm1\}^n$23

and the positive half

$x\in\{\pm1\}^n$24

After a small perturbation, the sets $x\in\{\pm1\}^n$25 form a partition of the cube (Sinha, 11 Sep 2025).

The same paper introduces the level-1 Fourier weight

$x\in\{\pm1\}^n$26

for Boolean functions $x\in\{\pm1\}^n$27, and proves the bound

$x\in\{\pm1\}^n$28

This reformulates the optimization problem in terms of a partition of the cube whose cells have large level-1 Fourier mass (Sinha, 11 Sep 2025).

The structural stability theorem in (Sinha, 11 Sep 2025) states that if $x\in\{\pm1\}^n$29 achieves the asymptotic maximum $x\in\{\pm1\}^n$30, then the cells $x\in\{\pm1\}^n$31 all have nearly equal volume,

$x\in\{\pm1\}^n$32

each row vector $x\in\{\pm1\}^n$33 almost coincides with the normalized centroid of $x\in\{\pm1\}^n$34, and for all but $x\in\{\pm1\}^n$35 indices $x\in\{\pm1\}^n$36, the set $x\in\{\pm1\}^n$37 is nearly isoperimetrically extremal for level-1 weight: $x\in\{\pm1\}^n$38 The paper summarizes this by saying that asymptotic extremizers induce “nearly uniform, centroidal Voronoi tessellations” of the cube (Sinha, 11 Sep 2025).

The same work also recasts the problem through a high-dimensional central limit theorem. If the rows are rescaled to Euclidean norm $x\in\{\pm1\}^n$39 via $x\in\{\pm1\}^n$40, and $x\in\{\pm1\}^n$41 are iid Rademacher variables, then

$x\in\{\pm1\}^n$42

and $x\in\{\pm1\}^n$43. Under mild nondegeneracy of $x\in\{\pm1\}^n$44 and bounded-entry hypotheses, a high-dimensional CLT gives

$x\in\{\pm1\}^n$45

so the extremal problem is closely tied to Gaussian maxima (Sinha, 11 Sep 2025).

6. Deterministic near-extremizers and open questions

The Gaussian viewpoint leads to sharper asymptotics and new deterministic constructions. For $x\in\{\pm1\}^n$46 with iid standard-Normal coordinates,

$x\in\{\pm1\}^n$47

and comparison theorems imply that among all covariance matrices with unit diagonal, the identity gives the largest $x\in\{\pm1\}^n$48 (Sinha, 11 Sep 2025).

For random sign matrices, the Gram off-diagonals satisfy

$x\in\{\pm1\}^n$49

and the CLT analysis yields

$x\in\{\pm1\}^n$50

with high probability (Sinha, 11 Sep 2025). This refines the earlier statement that random iid $x\in\{\pm1\}^n$51 matrices attain the optimal leading-order rate (Steinerberger, 2024).

The same paper constructs deterministic “Orthonormal Almost-Hadamard” matrices. One truncates a Hadamard matrix of order $x\in\{\pm1\}^n$52 to its top-left $x\in\{\pm1\}^n$53 block $x\in\{\pm1\}^n$54, normalizes by $x\in\{\pm1\}^n$55, and applies QR to obtain $x\in\{\pm1\}^n$56 with $x\in\{\pm1\}^n$57. Since all entries of $x\in\{\pm1\}^n$58 are $x\in\{\pm1\}^n$59, the CLT applies with $x\in\{\pm1\}^n$60, and

$x\in\{\pm1\}^n$61

Under Hadamard’s conjecture, this gives for all sufficiently large $x\in\{\pm1\}^n$62

$x\in\{\pm1\}^n$63

and the paper states that such matrices exist for infinitely many $x\in\{\pm1\}^n$64 unconditionally (Sinha, 11 Sep 2025).

Several open problems remain explicit in the literature. One is whether the gap between explicit constructions and the asymptotic optimum can be fully closed. Another is whether the lifting matrices are optimal when $x\in\{\pm1\}^n$65. A third is the exact determination of $x\in\{\pm1\}^n$66 for $x\in\{\pm1\}^n$67, where only highly structured candidates are currently available. The 2024 structural paper also asks about non-square and $x\in\{\pm1\}^n$68-variants, noting that only $x\in\{\pm1\}^n$69 are fully understood (Albors et al., 2024). These questions indicate that the problem now lies at the intersection of explicit combinatorial design, extremal Fourier analysis on the cube, and Gaussian comparison theory.