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The cosine measure of a function at a point

T0 review · 2 major / 3 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Every function and point carry a threshold — the cosine measure of its non-descent directions — and any search set whose cosine measure exceeds it must contain a descent direction, even without differentiability or a nonzero gradient.

desk verdict Sound new concept, fixable proof gap in Algorithm 2's theorem, useful l1 closed form; accept with revisions. read the letter →

arxiv 2608.07716 v1 pith:ZGDYTR4W submitted 2026-08-07 math.OC

Warren Hare , Gabriel Jarry-Bolduc , Chayne Planiden This is my paper · ORCID
classification math.OC MSC 90C5690C3049J5252A20
keywords cosinemeasuredescentdirectionsinfinitesetsoffinite-maxfunctionℓ1normnonsmoothoptimizationsecond-orderconeprogrampositivespanning
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper introduces the cosine measure of a function at a point, $\mathrm{cm}(f;\mathbf{x})$, defined as the cosine measure of the set of all unit non-descent directions of $f$ at $\mathbf{x}$. Its central theorem states that any set of search directions whose cosine measure exceeds $\mathrm{cm}(f;\mathbf{x})$ is guaranteed to contain a descent direction of $f$ at $\mathbf{x}$ — a guarantee that survives nondifferentiable functions and points where the gradient vanishes. The authors show by explicit examples that at a zero-gradient point the cosine measure of a smooth function can take any value in $[-1,1]$, so the threshold genuinely quantifies how hard a point is for direction-based descent. Algorithms compute the threshold exactly through second-order cone programs, including for infinite sets of directions; for finite-max functions the computation reduces to one SOCP built from active gradients, and for the $\ell^1$ norm it yields the closed formula $\mathrm{cm}(\lVert \mathbf{x}\rVert_1;\mathbf{x}_0)=\sqrt{m/n}$, where $m$ is the number of zero entries of $\mathbf{x}_0$. If the paper is right, direct-search methods get a per-point, per-function budget: a polling set whose cosine measure beats this value is provably assured of a descent direction, even where classical smooth analysis says nothing.

What carries the argument

The central object is the set $D^c(f;\mathbf{x})$ of unit non-descent directions of $f$ at $\mathbf{x}$, together with its cosine measure $\mathrm{cm}(f;\mathbf{x}) = \mathrm{cm}(D^c) = \min_{\lVert u\rVert=1} \sup_{d\in D^c} u^{\top}d$. The load-bearing identity is the threshold theorem: $\mathrm{cm}(S) > \mathrm{cm}(f;\mathbf{x})$ forces $S$ to contain a descent direction, with the proof running through a separating vector $u \in \mathrm{cV}(f;\mathbf{x})$ and the observation that a maximizer of $u^{\top}d$ over $S$ cannot lie in $D^c$. For computation, the mechanism is a second-order cone program: in the nonpositive case, $\mathrm{cm}(D^c) = \min\{t : \lVert u\rVert \le 1,\ u^{\top}d \le t \text{ for all } d\in S\}$, using the exactness of the convex relaxation when the cosine measure is nonpositive; in the positive case, the spherical complement of $\mathrm{cl}\,D^c$ is written as a disjoint union of interiors of pointed polyhedral cones $\mathrm{pspan}(S_i)$, and for each cone the algorithm computes its sine vector $u_i^*$ — the unit vector inside the cone farthest in angle from every boundary facet — via the program $\max\{t : \lVert u\rVert \le 1,\ p^{\top}u \ge t \text{ for all facet normals } p\}$, from which $\mathrm{cm}(D^c) = \min_i \sqrt{1-(t_i^*)^2}$. For finite-max functions, the facet normals of the relevant cone are exactly the normalized negative active gradients, collapsing the computation to a single SOCP; the sine vector of a cone coincides with the incenter of a proper cone.

What would settle it

Take a pointed polyhedral cone in $\mathbb{R}^3$ with facet normals $\{p_1,\dots,p_m\}$, solve the SOCP of Algorithm 2 for its sine vector $u^*$, and compare $\sqrt{1-(t^*)^2}$ with $\max_{d \in \partial C \cap \mathbb{S}^2} (u^*)^{\top}d$ computed by dense boundary sampling; if the two differ for any cone whose optimal vector's projection onto its nearest facet falls outside that facet, the identity behind Theorem 29 is refuted. A direct geometric check is simpler: for the computed $u^*$, verify that its orthogonal projection onto every facet hyperplane realizes the maximum of $(u^*)^{\top}d$ over the cone's boundary only for facets containing that projection — if the projection is outside the facet for the minimizing facet, the formula is unsupported.

Watch

Extended reading notes

Core claim

The central discovery is that every function $f$ and point $\mathbf{x}$ carry a number $\mathrm{cm}(f;\mathbf{x})$ — the cosine measure of the function at the point — equal to the cosine measure of the set $D^c(f;\mathbf{x})$ of all unit non-descent directions of $f$ at $\mathbf{x}$. Theorem 10 shows this number is a sharp threshold: every closed set $S$ of unit vectors with $\mathrm{cm}(S) > \mathrm{cm}(f;\mathbf{x})$ contains a descent direction of $f$ at $\mathbf{x}$, and the same holds for every orthogonal rotation of $S$; the threshold cannot be improved, because the set $D^c$ itself has cosine measure exactly $\mathrm{cm}(f;\mathbf{x})$ and contains no descent direction. The paper demonstrates that equality cases are richly attainable: at a point where $\nabla f(\mathbf{x}) = \mathbf{0}$, the value $\mathrm{cm}(f;\mathbf{x})$ can be anything in $[-1,1]$, so a vanishing gradient imposes no restriction on the threshold. It gives deterministic algorithms that compute $\mathrm{cm}(f;\mathbf{x})$ exactly — a second-order cone program when the closure of $D^c$ lies in a closed half-space, and a per-cone sine-measure computation over the facet normals of pointed polyhedral cones otherwise — and applies them to finite-max functions, where the non-descent directions are governed by the active gradients. For the $\ell^1$ norm this yields the closed form $\mathrm{cm}(\lVert\mathbf{x}\rVert_1;\mathbf{x}_0) = \sqrt{m/n}$, where $m$ is the number of zero entries of $\mathbf{x}_0$, with the consequence that any set $E$ of directions satisfying $\mathrm{cm}(E) > \sqrt{m/n}$ must contain a descent direction of the $\ell^1$ norm at $\mathbf{x}_0$.

Load-bearing premise

The load-bearing premise is a geometric assertion the proof leaves open in the positive-cosine case: that the direction most deeply hidden inside each pointed cone of search directions always sees its widest angle to the boundary at a facet whose containing hyperplane it projects onto from inside, so the projection never lands outside the cone; if that fails for some cone, the algorithm's returned value for the cosine measure is not established.

Editorial extensions

If this is right

  • Any direct-search polling set whose cosine measure exceeds $\mathrm{cm}(f;\mathbf{x})$ is guaranteed to contain a descent direction of $f$ at $\mathbf{x}$, with the guarantee now valid for nondifferentiable functions and zero-gradient points, not only for $\mathcal{C}^1$ functions with nonzero gradient.
  • The threshold is sharp: because $D^c(f;\mathbf{x})$ itself has cosine measure exactly $\mathrm{cm}(f;\mathbf{x})$ and contains no descent direction, no condition based only on the cosine measure of a set can force a descent direction below this value; when $\mathrm{cm}(f;\mathbf{x}) > -1$, a set can always be built at equality that fails to descend.
  • For the $\ell^1$ norm at a point with $m$ zero entries, any set $E$ of unit directions with $\mathrm{cm}(E) > \sqrt{m/n}$ must contain a descent direction at that point, giving a closed-form, coordinate-readable condition.
  • A vanishing gradient carries no information about the threshold: smooth functions at zero-gradient points realize every value of $\mathrm{cm}(f;\mathbf{x})$ in $[-1,1]$, and the value $0$ occurs exactly when the non-descent directions fit in some closed half-space.
  • The cosine measure of infinite sets of directions becomes computable exactly whenever the closure of the non-descent set is a finite union of finitely generated cones: one SOCP in the half-space case, one SOCP per pointed cone in the positive case.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • I read $\mathrm{cm}(f;\mathbf{x})$ as a per-point measure of descent difficulty: tabulating it for other standard nonsmooth functions — maximum eigenvalue, nuclear norm, hinge loss, $\ell^1$ of $A\mathbf{x}-\mathbf{b}$ — along the lines of Theorem 34 would produce a practical difficulty map for direction-based methods (the paper lists these functions as future work, not as a difficulty map).
  • The threshold suggests an adaptive polling strategy the paper does not develop: a derivative-free method could estimate $\mathrm{cm}(f;\mathbf{x})$ at the current iterate and construct a polling set whose cosine measure provably exceeds it, turning a static guarantee into a per-iteration dynamic one.
  • Because the sine vector of a cone is the incenter of a proper cone, existing incenter algorithms and bounds could supply estimates or upper bounds for $\mathrm{cm}(f;\mathbf{x})$ in high dimensions, where enumerating cone facets is exponential.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 3 minor

Summary. The paper introduces the cosine measure of a function at a point, cm(f;x), defined as the cosine measure of the set D^c of all unit non-descent directions of f at x. The central result, Theorem 10, states that any closed set S of unit vectors with cm(S)>cm(f;x) must contain a descent direction of f at x. The paper proves basic properties of cm(f;x), gives examples showing that every value in [-1,1] can occur when the gradient vanishes, and develops three algorithms: Algorithm 1 for the case where cl D^c is a finite union of polyhedral cones contained in a closed half-space, Algorithm 2 for the complementary positive case via the spherical complement, and Algorithm 3 specialized to finite-max functions. It closes with a closed-form expression for the cosine measure of the l1 norm at a point with m zero coordinates, cm = sqrt(m/n).

Significance. If the results are correct, cm(f;x) provides a natural worst-case threshold for guaranteeing a descent direction in settings where the classical gradient condition fails, and it extends cosine-measure theory from finite positive spanning sets to infinite and nonsmooth directional sets. The l1-norm formula in Theorem 34 is elegant and immediately usable in derivative-free optimization analyses. The paper contains several strengths: Theorem 10 has a short, correct proof; the examples in Section 3.1 collectively demonstrate attainability of all values in [-1,1]; the algorithms are deterministic and reduce to second-order cone programs; and the exposition is generally clear. The main weakness is that the correctness proof of Algorithm 2 contains an unproven geometric assertion, and both Algorithms 1 and 2 rely on an unpublished companion result. These issues are fixable but currently block verification of a load-bearing part of the paper.

major comments (2)
  1. [Section 4.2, Theorem 29] The proof of Theorem 29 asserts, without proof, that max_{d in delta-pspan(S_i)} d^T u_i^* equals max_{j_i} ||Proj_{H_{j_i}} u_i^*||, and that the maximum is attained at a facet realizing the minimal distance from u_i^*, for which the projection lies in the cone. This assertion is load-bearing: if the projection onto the closest facet fell outside that facet, the maximum could be attained on a lower-dimensional face and the formula cm((S_i^o)^c) = max_{p in P_i} sqrt(1-(p^T u_i^*)^2) would not follow. The text only notes that the projection 'need not lie in pspan(S_i)' and does not prove the required containment for the optimal sine vector. Please insert the missing argument: for an active normal p_j with p_j^T u_i^* = t_i^* = min_{p in P_i} p^T u_i^*, the vector q = u_i^* - (u_i^{*T} p_j) p_j satisfies p_l^T q = p_l^T u_i^* - t_i^*(p_j^T p_l) >= t_i^*(1 - p_j^T p_l) >= 0 for every p_l in P_i, so q lies in the cone and on the boundary; this establishes the asserted equality and the correctness of the algorithm's formula.
  2. [Section 4.1 and 4.2] The correctness proofs of Algorithms 1 and 2 depend on Lemma 33 of reference [2], which is cited as an unpublished manuscript with overlapping authorship. This lemma is used in the proof of Theorem 25 to justify the exactness of the convex relaxation for nonpositive cosine measure, and it is invoked again at the end of Theorem 29 to justify the convex relaxation in the positive case. Because this lemma is exactly what licenses replacing the nonconvex cosine-measure problem by the SOCPs in Steps (1.1) and (1.2), the paper should either state and prove the needed lemma in a self-contained appendix or give the precise statement of Lemma 33 and explicitly identify the dependency. As written, the reader cannot verify a load-bearing step without consulting a separate unpublished paper.
minor comments (3)
  1. [Section 4.2] The examples in Section 5 (Examples 33 and 34) exercise only Algorithm 3, which is a simplified version of Algorithm 2 for a single cone. A small hand-worked or numerical instance of Algorithm 2 with k > 1 cones (for example, the configuration of Example 11) would materially help validate Step 2 and the assumption that pspan(S_i) cap pspan(S_j) = {0}.
  2. [Definition 1 and Section 4] Definition 1 uses the same symbol for the positive span and for the normalized positive span, and Algorithms 1, 2, and Theorem 21 rely on the normalized version. Since the two notions are used interchangeably in the displayed text, please introduce a distinct symbol for the normalized cone (for example, overline-pspan) to avoid ambiguity.
  3. [Throughout] There are several typographical slips that should be corrected in a revision: 'the the cosine measure' in the abstract, 'authours' in the acknowledgments, 'particulary' and 'in particualar' in Section 5, and 'The set D^c for f_alpha at x_0 = 0 is represented by the red spherical cap' in Example 16 where the figure is not included in the text.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: cm(f;x) is transparently defined as cm(D^c), and Theorem 10 follows by a direct argument; the algorithm proofs cite a companion-paper lemma and contain an unproven projection assertion, but neither reduces the central claim to its own inputs.

full rationale

The paper defines cm(f;x) explicitly as cm(D^c), the cosine measure of the set of all unit non-descent directions, with the cosine vector set as the associated argmin (Definition 8). Theorem 10 is then a short consequence: for u in cV(f;x), cm(S) > cm(f;x) = sup_{d in D^c} u^T d forces some v in S to satisfy u^T v > sup_{d in D^c} u^T d, so v is not in D^c and hence is a descent direction. This is a derivation from the definition, not a hidden equivalence; there are no fitted parameters, no predictions, and no empirical inputs. The examples and the closed-form formula for the l1 norm are computed from the same definition and are externally checkable. The only load-bearing citation to overlapping-author prior work is Lemma 33 of [2], invoked in the correctness proofs of Algorithms 1 and 2 (Theorem 25; Theorem 29); however, [2] is a parameter-free prior result whose assumptions do not include the present target result, so under the review rules it counts as real evidence and does not raise the circularity score. Separately, Section 4.2, Theorem 29, contains an unproven assertion about projections onto facet hyperplanes: the text states that the maximum of ||Proj_{H_{j_i}} u^*_i|| is attained at a hyperplane realizing the minimal distance from u^*_i, for which the projection lies in pspan(S_i), without proof. This is a correctness gap that should be repaired before acceptance, but it is not a circularity. Overall, the central derivation chain is self-contained and non-circular, with no step reducing to its own input.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no fitted free parameters and no invented physical entities. It relies on structural domain assumptions: clD^c or D^o must be a finite union of polyhedral cones for the algorithms, and the finite-max specialization assumes active gradients are nonzero. It also imports exactness of a convex relaxation from the companion paper [2]. The central threshold definition itself is self-contained.

assumptions (4)
  • domain assumption The set of non-descent directions of a function at a point is a union of finitely many polyhedral cones (Eq. (1)), or its spherical complement is a union of interiors of polyhedral cones with pairwise-trivial intersection (Eq. (5)).
    This structural assumption is stated in Section 4 and is required for Algorithms 1 and 2 to be applicable.
  • standard math The convex relaxation of the cosine measure problem is exact when the cosine measure is nonpositive ([2, Lemma 33]).
    Used in the proofs of Theorems 25 and 29 to replace the unit-sphere constraint by the unit-ball constraint.
  • domain assumption For finite-max functions, the Clarke subdifferential is the convex hull of active gradients and D^c is the union over active indices of halfspaces {d : grad f_i(x)^T d >= 0}.
    Used in Section 5 to specialize Algorithm 2 to finite-max functions; follows from known nonsmooth analysis.
  • standard math Orthogonal transformations preserve cosine measure (Lemma 9, from [34]).
    Used in Theorem 10 to guarantee rotated sets also contain a descent direction.

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Pith. "Pith review of The cosine measure of a function at a point." pith.science (2026). https://pith.science/paper/ZGDYTR4W

@misc{pith2026260807716,
  author       = {Pith},
  title        = {Pith review of: The cosine measure of a function at a point},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZGDYTR4W}},
  note         = {Machine review of arXiv:2608.07716}
}
abstract

The cosine measure of a set of vectors in $\mathbb{R}^n$ measures how well the set covers all directions in $\mathbb{R}^n$. It identifies the direction furthest, in angle, from the set. It is used in the convergence theory of various optimization algorithms, but also highlights interesting geometric properties of sets. For example, the cosine measure of a set $S$ is greater than zero if and only if given any $\mathcal{C}^1$ function $f$ at a point $\mathbf{x}$ where the gradient is nonzero, $S$ must contain a descent direction of $f$ at $\mathbf{x}$. In this paper, we examine the question of what can be said when the function $f$ is non-differentiable or if it has a gradient equal to the zero vector. To examine these cases, we introduce the novel concept of the {\em cosine measure of a function} at a point. This value provides an infimum on the value of the cosine measure that a set of vectors requires to guarantee it contains a descent direction of the function at the point of interest. We present mathematical theory around this concept, including examples showing that the cosine measure of a smooth function can have any value in $[-1,1]$. We further present algorithms to compute the cosine measure of a function, and examples demonstrating the algorithm on smooth and nonsmooth functions. These results also shed light on the the cosine measure of infinite sets and nonconvex cones.

Figures

Figures reproduced from arXiv: 2608.07716 by the authors.

Figure 1
Figure 1. Any rotation or reflection of the set S contains a descent direction of f at x0 = 0, but cm(S) = 1/2 < 1/ √ 2 = cm(f; 0). The two green spherical caps represent the descent directions of f at x0. From the previous example, we conclude that it is possible to create a set S with lower cosine measure than cm(f; x) and identify a descent direction even when the set is rotated. However, it is also possible to create a se… view at source ↗
Figure 2
Figure 2. The set Dc for fα at x0 = 0 is represented by the red spherical cap. On the left, α = 0.5, and on the right α = 10. Note that fα is differentiable at x0 = 0. The contour lines at f(x) = 0 are given by the equations x2 = √ α|x1|. As a vector-valued function, the contour lines at f(x) = 0 are L1(t) = h √ 1 1+α q α 1+α i⊤ t and L2(t) = h √−1 1+α q α 1+α i⊤ t, where t ≥ 0. We find that the set Dc is pspan h √ 1 1+α q α… view at source ↗
Figure 3
Figure 3. On the left, the set {(x1, x2) : 0 < x1, −x1 < x2, ∥x∥ = 1} is represented by one spherical cap. The cosine vector u ∗ is found using Algorithm 2. On the right, the set is decomposed into two overlapping spherical caps. A vector u ∗ is computed for each spherical cap. This yields an incorrect (overestimated) value of the cosine measure. Examining the special cases (Step 0), we notice that if Algorithm 2 is applied t… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Graph and the descent directions of F = max{x1 + x2, −x1 + x2} at x0 = 0 (represented by the green spherical cap). The active set of indices at x0 = 0 is I(x0) = {1, 2}, so the normalized active negative gradient set is P =  p1 = 1 √ 2  −1 −1  , p2 = 1 √ 2  1 −1 …
Figure 5
Figure 5. Figure 5: The cosine vector u ∗ of F at x0 = 0 is not equal to the steepest descent direction usd Our second example considers the common regularization function ℓ1. Recall, in R n, ∥x∥1 = Xn i=1 |xi | = max σ∈{−1,1}n σ ⊤x, where the notation {−1, 1} n means that each entry of t…

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Pith tools

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