Near-Optimal Bounds for Sketching the Schatten--1 Norm

Abstract: Let $k_ε(n)$ be the smallest number of real linear measurements needed by a randomized, oblivious sketch that estimates the nuclear norm of every fixed real $n\times n$ matrix within a factor $1\pmε$, with probability at least $2/3$. For every fixed $0<ε<1$, the proved result is $$\frac{n^2}{(\log n)^{A_ε}} \;\le\; k_ε(n) \;\le\; C_ε\frac{n^{2{\log\log(e^e} n)}^2}{\log(e n)}$$ for all sufficiently large $n$, where $A_ε,C_ε$ depend only on $ε$. Previously, the best bounds for general linear sketches were $Ω(n)$ and the trivial $O(n<sup>2)$ upper bound (Li, Nguyen, Woodruff, 2019). The theorem therefore nearly resolves the open measurement-complexity question left by that work: the displayed lower and upper bounds are tight up to polylogarithmic factors. In particular, the complexity is $n<sup>{2-o(1)}$, and for every fixed $c&gt;0$, $O(n<sup>{2-c})$ measurements are impossible. The upper bound is obtained by a fixed Gaussian sketch whose decoder combines implicit low-rank recovery with moment estimation on a high-stable-rank residual. The lower bound constructs moment-matched spectra, randomizes their singular vectors, and compares every low-dimensional observation through an odd-order tensor estimate and a Fisher-information path argument.

•  The paper establishes the near-optimal bounds for estimating the nuclear norm of an arbitrary real $n  imes n$ matrix using oblivious randomized linear sketches, showing that $n^{2-o(1)}$ exact real measurements are necessary and sufficient.
• The lower bound is established through a distributional, decoder-independent proof, demonstrating that it is NP-hard problem to to distinguish between indistinguishable nuclear norms based on measured distributions.
• Using an oblivious Gaussian sketch, an adaptive spectral decomposition decoder achieves record-breaking upper bounds of $O(n^2-log(log n)^2/)$ and thus ensuring data fidelity with high probability. 

Problem and principal result

The paper studies the measurement complexity of estimating the nuclear norm, or Schatten–1 norm, of an arbitrary real $n \times n$ matrix using an oblivious randomized linear sketch. A sketch consists of a random linear map from the $n^2$ matrix entries to $k$ real measurements, followed by an arbitrary measurable decoder. The guarantee is pointwise over matrices: for every fixed matrix $A$, the decoder must return a $(1 \pm \varepsilon)$ approximation to $\|A\|_{S_1}$ with constant probability. The randomness is therefore allowed to depend on neither the input nor its distribution.

This model is substantially stronger than bilinear sketches and distinct from finite-bit streaming space. The measurements are exact real numbers, so $k$ counts data-dependent real linear measurements rather than bits. The distinction is essential: a lower bound on the number of exact measurements does not automatically imply an equivalent lower bound for arbitrary finite-memory streaming algorithms.

The main theorem establishes, for every fixed $0 < \varepsilon < 1$, constants $A_\varepsilon,C_\varepsilon>0$ such that, for sufficiently large $n$,

$n^2$0

Thus the measurement complexity is $n^2$1. In particular, no sketch using $n^2$2 measurements exists for any fixed $n^2$3. The result closes the polynomial gap between the previously known $n^2$4 lower bound and the trivial $n^2$5 upper bound, leaving only polylogarithmic uncertainty. The paper consequently identifies the correct polynomial order of the general-linear-sketch problem, although it does not determine the optimal logarithmic factor (2608.22247).

The contrast with positive semidefinite inputs is instructive. If $n^2$6, then $n^2$7, so one linear measurement suffices. The near-quadratic complexity therefore arises from the unknown and independent left and right singular directions of a general matrix, not merely from the operation of summing singular values.

Lower bound through indistinguishable randomized spectra

The lower bound is distributional and decoder-independent. The proof reduces nuclear-norm estimation to binary hypothesis testing using a Yao–Le Cam argument. For a fixed rank-$n^2$8 linear measurement map, the authors construct two input distributions whose nuclear norms occupy disjoint relative-error intervals but whose measured distributions have small total variation distance. Any successful estimator would distinguish the two hypotheses, contradicting the statistical indistinguishability of the measurements.

A technical point is that an arbitrary rank-$n^2$9 measurement map can be reduced to an orthogonal projection onto a $k$0-dimensional subspace of the $k$1-dimensional matrix space. If $k$2 has rank $k$3, it factors as $k$4, where $k$5 has orthonormal rows and $k$6 is injective. Hence $k$7 contains all information in $k$8, and it suffices to analyze row coisometries and their associated orthogonal projections.

Moment matching versus the nuclear norm

The hard instances are built at the level of squared singular values. The construction produces two scalar probability laws $k$9 and $A$0 supported on a polynomial-size interval such that

• their moments agree through degree $A$1;
• their first moments are normalized to one; and
• their square-root expectations differ by a constant depending only on $A$2.

The square-root expectation is the relevant functional because if $A$3 is a squared singular value, then $A$4 is a singular value. Therefore matching the moments of $A$5 does not determine the quantity $A$6.

The scalar construction uses alternating Lagrange-interpolation weights on nodes of the form $A$7, for $A$8. The signed weights annihilate every polynomial of degree at most $A$9. Splitting the signed measure into positive and negative parts gives two probability laws with identical moments up to degree $(1 \pm \varepsilon)$0. By taking $(1 \pm \varepsilon)$1 sufficiently large, one law concentrates most of its mass near zero while the other concentrates mass near one, producing a separated square-root expectation.

This construction is converted into actual $(1 \pm \varepsilon)$2-point empirical spectra. That conversion is nontrivial because arbitrary probability weights need not be integer multiples of $(1 \pm \varepsilon)$3. The paper reserves small groups of control particles and adjusts them to enforce exact equality of the first $(1 \pm \varepsilon)$4 empirical moments. Chebyshev polynomials provide a well-conditioned coordinate system for these corrections, while a contraction argument controls the nonlinear adjustment.

More importantly, the authors construct a continuous path between the two empirical spectra along which every moment through degree $(1 \pm \varepsilon)$5 remains constant at every intermediate time. This pathwise condition is necessary: endpoint moment matching alone would not control the Fisher information accumulated along a path whose low moments vary substantially in the interior. The path length is bounded by

$(1 \pm \varepsilon)$6

where $(1 \pm \varepsilon)$7 bounds the squared singular values and $(1 \pm \varepsilon)$8 is the path-correction tolerance.

Random singular vectors and Gaussian regularization

The empirical diagonal spectra are embedded into random matrices

$(1 \pm \varepsilon)$9

where $\|A\|_{S_1}$0 and $\|A\|_{S_1}$1 are independent Haar-distributed orthogonal matrices. The nuclear norms of $\|A\|_{S_1}$2 and $\|A\|_{S_1}$3 are deterministic and separated, while their low-order tensor moments agree as a consequence of the matched spectral moments.

The orbit distributions of $\|A\|_{S_1}$4 are singular with respect to Lebesgue measure, which prevents direct use of density-based information arguments. The paper therefore adds common Gaussian noise:

$\|A\|_{S_1}$5

The noise is chosen small enough that the nuclear-norm gap survives with high probability. At the same time, under any orthonormal-row measurement map $\|A\|_{S_1}$6, the projected noise remains a standard Gaussian vector:

$\|A\|_{S_1}$7

This regularization produces smooth positive densities while preserving the statistical structure needed for the lower bound.

Tensor cancellation and the odd-order obstruction

The central analytic mechanism is repeated integration by parts on the orthogonal groups. If the first $\|A\|_{S_1}$8 squared-singular-value moments remain constant along the path, then all tensor moments of the randomized matrices through degree $\|A\|_{S_1}$9 are constant. The derivative of the measured density consequently has no contribution at these orders.

The first potentially nonzero term appears at the odd order

$k$0

The paper proves an odd-order two-sided rotation inequality. For every rank-$k$1 projection $k$2 on the matrix space and every order-$k$3 tensor $k$4,

$k$5

provided $k$6.

The exponent $k$7 is the decisive feature. It reflects the fact that an odd tensor order necessarily leaves at least one row–column path in the contraction graph induced by the measurement projection. The proof decomposes tensors into trace-free components indexed by partial matchings, applies orthogonal Weingarten calculus to each block, and bounds the resulting measurement contractions by a graph calculation. The trace-free decomposition is quantitatively stable because distinct pairing blocks have overlap at most $k$8 when the number of blocks is sufficiently smaller than $k$9.

The tensor estimate is combined with a Fisher–Rao path bound. If $0 < \varepsilon < 1$0 denotes the law of the measured, Gaussian-smoothed matrix along the spectral path, then

$0 < \varepsilon < 1$1

where $0 < \varepsilon < 1$2 is the Fisher information along the path. The score is represented exactly using Hermite tensors, without a Taylor remainder. The resulting estimate has the form

$0 < \varepsilon < 1$3

Taking

$0 < \varepsilon < 1$4

and accounting for the polynomial growth of $0 < \varepsilon < 1$5 and the exponential-in-$0 < \varepsilon < 1$6 losses yields indistinguishability whenever

$0 < \varepsilon < 1$7

Le Cam’s identity then implies that no decoder can estimate the nuclear norm with the required success probability. This lower bound applies uniformly to every linear measurement subspace and every measurable decoder, rather than only to a particular sketch architecture.

The proof also identifies its own quantitative bottleneck. At $0 < \varepsilon < 1$8 for constant $0 < \varepsilon < 1$9, the contraction term would require $A_\varepsilon,C_\varepsilon>0$0 of order $A_\varepsilon,C_\varepsilon>0$1, but the tensor comparison incurs losses of order $A_\varepsilon,C_\varepsilon>0$2. The authors explicitly state that this limitation does not establish a quadratic lower bound; it only limits the strength of the present moment-matching and tensor-comparison method.

Upper bound through adaptive spectral decomposition

The upper bound uses a fixed Gaussian sketch whose decoder performs a data-dependent spectral decomposition. The sketch stores four independent Gaussian products,

$A_\varepsilon,C_\varepsilon>0$3

All maps are sampled before seeing $A_\varepsilon,C_\varepsilon>0$4, so the sketch remains oblivious and linear. The decoder, however, can use the stored products to select a cutoff after observing the data.

The algorithm separates the singular spectrum into a head and a tail. The head contains a controlled number of dominant singular directions, while the tail is handled by moment estimation only after it has been certified to have sufficiently large stable rank.

A multiscale head–tail dichotomy

For a cutoff $A_\varepsilon,C_\varepsilon>0$5, let $A_\varepsilon,C_\varepsilon>0$6 denote the nuclear norm of the singular-value tail beyond rank $A_\varepsilon,C_\varepsilon>0$7, and let $A_\varepsilon,C_\varepsilon>0$8 denote its Frobenius norm. The decoder examines cutoffs in blocks of size $A_\varepsilon,C_\varepsilon>0$9. A deterministic dichotomy shows that at one of these cutoffs, either

1. the tail nuclear norm is at most an $n$0 fraction of $n$1, or
2. the exact residual has stable rank at least a constant multiple of $n$2.

The proof uses the contrapositive. If every residual has stable rank below $n$3, then the singular value at the beginning of successive blocks decreases geometrically. After sufficiently many inspected blocks, the remaining nuclear norm must therefore be negligible.

This dichotomy is the structural basis for the upper bound. Concentrated tails cannot be estimated reliably from a small number of spectral moments, so they are absorbed into the head or discarded when already negligible. Spread-out tails have bounded normalized squared singular values and are suitable for polynomial approximation.

Recovering an implicit head

The first Gaussian block $n$4 identifies approximate top right-singular subspaces. A covariance concentration theorem with an effective-rank dependence gives, simultaneously over all candidate cutoffs, an approximation of $n$5 by $n$6 with an additive regularization term controlled by the Frobenius mass of the corresponding exact tail.

If $n$7 is the top-$n$8 right singular subspace extracted from $n$9 and $n^2$00, the residual

$n^2$01

satisfies the relative nuclear-norm comparison

$n^2$02

The corresponding head and residual norms are approximately additive:

$n^2$03

The use of a total-relative error, rather than an error relative to the residual tail, is important. A residual-relative guarantee would require substantially more measurements when the tail is small.

The second Gaussian block $n^2$04 estimates the Frobenius norm of every candidate residual. Together with the next singular value of $n^2$05, this permits a stable-rank certificate. Passing the certificate implies

$n^2$06

Conversely, an exact residual with stable rank at least $n^2$07 passes the certificate. Thus, if no candidate is certified, the multiscale dichotomy guarantees that the final residual has negligible nuclear norm.

The selected head is not explicitly reconstructed from $n^2$08. Instead, the independent block $n^2$09 is combined with the selected subspace through Gaussian regression:

$n^2$10

Since $n^2$11 is independent of the subspace-selection procedure, inverse-Wishart estimates imply

$n^2$12

Choosing $n^2$13 proportional to the largest candidate rank makes this error at most $n^2$14 with high probability. This step is a key use of independent sketch blocks: the subspace may be data-dependent, but the regression measurements remain fresh conditional on that subspace.

Estimating a stable-rank tail

For a certified residual $n^2$15, define normalized squared singular values

$n^2$16

The stable-rank certificate implies

$n^2$17

The residual nuclear norm can be expressed as

$n^2$18

The decoder therefore estimates the average square root of the normalized spectrum.

The square-root function is uniformly approximated on the bounded interval $n^2$19 by a degree-$n^2$20 polynomial. The paper uses a polynomial whose coefficients satisfy an exponential bound in $n^2$21 and whose approximation error is $n^2$22 after rescaling. The required degree is proportional to $n^2$23, hence to the block parameter $n^2$24.

The moments of the $n^2$25 are estimated from the fourth Gaussian block $n^2$26. If $n^2$27 denotes a suitably normalized row of $n^2$28, the decoder forms injective Gaussian cycle statistics. For degree $n^2$29,

$n^2$30

and these statistics are unbiased:

$n^2$31

After normalizing by $n^2$32, the decoder obtains estimates of the normalized moments $n^2$33. A variance theorem for Gaussian cycle statistics controls the required number of rows. Because the residual spectrum is bounded, the polynomial coefficients and moment errors can be balanced with degree

$n^2$34

and a tail sketch using

$n^2$35

real measurements.

The algorithm combines the estimated head and tail. With $n^2$36, the certified branch incurs total relative error at most $n^2$37, while the negligible branch incurs at most $n^2$38. The stated parameter choice gives an error at most $n^2$39, stronger than the required $n^2$40 guarantee, with success probability at least $n^2$41.

Optimization of the measurement count

The block parameter is chosen as

$n^2$42

The head-selection, covariance, certification, and regression blocks use

$n^2$43

measurements. The tail block uses $n^2$44 measurements up to polynomial factors in $n^2$45, and the chosen value of $n^2$46 makes this term no larger than the head cost. Consequently, the complete oblivious Gaussian sketch satisfies

$n^2$47

The result is an explicit subquadratic saving over storing all $n^2$48 entries, although the saving is only logarithmic up to iterated-logarithmic factors.

Relation to prior sketching and streaming results

The paper’s contribution depends strongly on maintaining the distinction between several computational models. Earlier work established an $n^2$49 lower bound for general exact-real linear sketches and an $n^2$50 upper bound obtained by retaining the entire matrix. Stronger lower bounds were known for restricted bilinear sketches and for finite-bit streaming algorithms on sparse bounded-integer inputs, but those statements do not imply the present theorem.

The upper bound is also not a standard bilinear sketch. It stores both left products such as $n^2$51 and $n^2$52 and right products such as $n^2$53, together with a second left product $n^2$54. The four-block architecture allows the decoder to select a subspace using one part of the sketch and then perform independent regression and moment estimation using the remaining parts.

The result differs qualitatively from vector $n^2$55 sketching. In vector problems, stable random projections yield dimension-independent sketch size for fixed relative accuracy. For matrices, singular values are nonlinear functions of the entries, and entry updates affect all singular directions simultaneously. The near-quadratic lower bound formalizes the difficulty created by jointly unknown left and right singular bases.

Limitations and open questions

The theorem leaves the logarithmic dependence unresolved. The lower bound is

$n^2$56

whereas the upper bound is

$n^2$57

The paper does not determine whether the upper-bound logarithmic saving is optimal, whether the lower-bound exponent can be improved, or whether iterated logarithms are intrinsic.

The dependence on $n^2$58 is also not optimized. The result fixes $n^2$59 before taking $n^2$60 to infinity. A joint theorem for $n^2$61 would require sharper tracking of the moment-matching order, smoothing scale, polynomial degree, and Gaussian sample sizes.

The exact-real model limits the interpretation in finite precision. The lower bound rules out small collections of exact linear counters on the full real input domain, but it does not establish an $n^2$62-bit lower bound for arbitrary streaming algorithms. Likewise, the Gaussian upper bound does not provide a finite-precision implementation: the paper does not bound the bits required to represent the random maps and measurements, nor does it analyze numerical stability of pseudoinverses, cutoff selection, or high-order cycle statistics.

Finally, the theorem is proved only for unrestricted real square matrices in the one-pass general-linear-sketch model. Rectangular and complex matrices, bilinear restrictions, row-order streams, sparse inputs, update time, decoding time, and finite-precision stability remain separate questions rather than direct consequences of the result.

Conclusion

The paper establishes that approximating the nuclear norm of an arbitrary real $n^2$63 matrix from an oblivious randomized linear sketch requires $n^2$64 exact real measurements. Its lower bound combines moment-matched spectra, random singular vectors, Gaussian smoothing, Fisher-information interpolation, and odd-order tensor contraction estimates. Its upper bound uses multiscale head selection, stable-rank certification, implicit subspace regression, and polynomial spectral-moment estimation. Together, these arguments reduce the previously polynomially unresolved complexity gap to polylogarithmic factors, while leaving the optimal logarithmic dependence and the finite-precision interpretation open (2608.22247).