Nonlinear reconciliation: Error reduction theorems
Authors: Lorenzo Nespoli
Anubhab Biswas
Roberto Rocchetta
Vasco Medici
Venue: Accepted by TMLR ()

Abstract: Forecast reconciliation, an ex-post technique applied to forecasts that must satisfy constraints, has been a prominent topic in the forecasting literature over the past two decades. Recently, several efforts have sought to extend reconciliation methods to the probabilistic settings. Nevertheless, formal theorems demonstrating error reduction in nonlinear contexts, analogous to those presented in Panagiotelis et al., (2021), are still lacking. This paper addresses that gap by establishing such theorems for various classes of nonlinear hypersurfaces and vector-valued functions. Specifically, we derive an exact analog of Theorem 3.1 from Panagiotelis et al., (2021) for hypersurfaces with constant-sign curvature. Additionally, we provide an error reduction theorem for the broader case of hypersurfaces with non-constant-sign curvature and for general manifolds with codimension > 1. To support reproducibility and practical adoption, we release a JAX-based Python package, JNLR, implementing the presented theorems and reconciliation procedures.

Published in Transactions on Machine Learning Research (01/2026)
Nonlinear reconciliation: Error reduction theorems
Lorenzo Nespoli
lorenzaoneepolkosssss...
ISAAC
SUPSI
Anubhab Biswas
anubhab muubhab.biswasSee Sisphs
ISAAC
SUPSI
Roberto Rocchetta
ISAAC
SUPSI
Vasco Medici
wusco.mecdiciisspssi...
ISAAC
SUPSI
Reviewed on OpenReview: https: // openreview. net/ forum? id= dXRWuogm33
Abstract
Forecast. reconciliation, an ex-post technique applied forecasts that must satisfy con- straints, has been a prominent topic the forccasting literature over the past two docades. Recently, several efforts have sought extend reconciliation methods to the probabilistic settings. Nevertheless, formal theorems demonstrating error reduction in nonlinear con- straints, analogous to those presented in Panagiotelis et al. (2021). are still lacking. This paper addresses that gap by establishing such theorems for various classes of nonlinear hy- persurfaces and vector-valued functions. Specifically, we derive an exact analog of Theorem 3.1 from Panagiotelis et al. (2021) for hypersurfaces with constant-sign curvature. Addi- tionally, we provide an error reduction theorem for the broader case of hypersurfaces with To support reproducibility and practical adoption, release JAX-based Python package, JNLR, implementing the presented theorems and reconciliation procedures.
non-constant-sign curvature and for general manifolds with codimension
1 Introduction
Forecast reconciliation is post-processing technique used to minimally correct original set of forecasts, ensuring they satisfy known structural constraints. A straightforward example is the case of additive signals, grouped into increasingly larger sets, such as the quantity of a given good sold at the regional, state, and national levels. In such cases, the forecasts must obey additive relationships, a setting typically referred to as hierarchical forecasting (HF) the literature.
Forecast reconciliation has become a widely adopted technique because it can be applied ex-post, indepen- dently of the forecasting model. This makes it model-agnostic method for enforcing known structural constraints forecasted signals. Reconciliation is typically motivated by two main considerations. An cristential reason, if forecasts not respecting the known constraints are unusable or of little value. In this case reconciling is not an option. The second whililarian reason: knowing the signals lie on a subman- ifold induces a bias in the production of the forecasts; this can lower the forecasting error. In the case of lincar constraints, Theorem
|Panagiotelis al.] (2021) shows that orthogonally projecting forccasts onto the consistent subspace reduces the overall root mean squared error (RMSE) across the hicrarchy. This refers to the RMSE of single forecast. vector at given time point, computed the square root of the

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average of its squared components (as opposed to the RMSE over an entire test set). However, when the constraints are nonlinear, such reduction is not always guaranteed: if reconciliation is performed solely to boost coouracyraather than to strictly enforce the constraints it can in fact lead to worse predictions. The ability to reliably estimate the probability of RMSE reduction under nonlinear constraints will significantly drive the adoption of reconciliation for improving lorecasting accuracy. In this paper, we present. theorems that establish conditions under which this RMSE reduction holds. In other words, these conditions answer the question: Should I expect an RMSE reduction when applying nonlinear reconciliation to the current forecasted point?
1.1 Linear reconciliation
Originally studied in industrial management as family-based forecasting Fliedner 2001 HF remains an active research topic, as evidenced by its inclusion in the 2020 M5 competition, where the organizers con- cluded that new methods capable of minimizing errors across all levels could offer significant accuracy gains [Makridakis et al. 2022). The simplest reconciliation strategy, bottom-up (Orcutt et al., 1968 Dunn et al,, 1976), forecasts only the most disaggregated series, aggregating them upward. However, bottom-level series often have low signal-to-noise ratios, limiting performance. Forecasting all series independently leads to incoherent results unless reconciled. A major step forward was presented in (Hyndman et al. 2011 1, where the authors proposed an optimal reconciliation based on generalize least square (GLSS.
Later refinements (Athanasopoulos et al. of base forecast errors. Later modification of the method include dropping the unbiasedness assumption [Ben Taieb et al. 2017 Di Modica et al.
2009
Wickramasuriya et al. 2019 proposed the use of covariance
2021), extension to temporal hierarchies, where series are aggregated over time (Athanasopoulos et al.) 2017 Yang et al. 2017 and learning dynamic hierarchies (Cini et al. 2020 2024). While point forecast reconciliation is well established, probabilistic reconciliation remained at first under-explored due to its complexity. Closed-form solutions exist only under strong assumptions (o.g., Gaussianity) (Corani et al. 2021), or errors' independence, when aggregation can be performed using convolutions. Other techniques based on sample reordering or copulas have been proposed (Jeon et al. 2019 Taieb et al., (021), though they require knowledge of the joint cumulative density function (CDF) at each level or rely on empirical approximations. A recent framework reconciles samples from joint but incoherent distributions via constrained optimization based on scoring rules (Panagiotclis ct al. 2023). outperforming copula methods but still assuming known joint CDF.
1.2 Non-linear reconciliation
The nonlinear reconciliation problem has been less studied in the context of forecasting. Since the nonlinear reconciliation problem doesn't. usually have an analytical, closed-form solution, its application less attrac- tive than its linear counterpart. A method that could be used to produce a set of coherent forecasts for nonlinear manifolds, even if the authors applied it only to linear hierarchics, has been proposed by SSpllotti et al. (202T), , and later applied in Rombouts et al. (2025), where nonlinear regressor is trained to produce forecasts for the independent signal starting from the independent forecasts of all the signals. However, this method needs to train a machine learning model, and one has no control on the kind of projection applied, since this is implicitly learned by the model. On the other side, in the context of state estimation, nonlinear reconciliation techniques are well known. At the intersection of forecasting and nonlinear state estimation, the forecast aided state estimation (FASE) method has been proposed in the context of power grid state estimation (Brown Do Coutto Filho& & De Souza 2009). In FASE the forecasting model is linear state-space, and the reconciliation step is equivalent to iterated extended Kalman filter. In appendix A.1 we show that the FASE objective function represents a special case of the nonlinear reconciliation problem we introduce in the next section. That is, the theory presented in this paper is also valid for FASE and its application to power flow state estimation and forecasting.
1.3 Contributions
The main contributions are the following:
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1. In section define the nonlinear reconciliation problem and algorithm for its solution. In appendix A. we show that this algorithm is a generalization of the FASE algorithm.
2. In section 8 we present theorems providing point-wise RMSE reduction conditions for the the non- linear reconciliation problem, for different classes of manifolds M implicitly defined by level sets of a function ʃ. We propose two theorems, | and 2 for constant sigu curvature manifolds and a third one, Theorem 3 providing a probabilistic reduction condition, which can be applied in the general case of M with codimension >I and without assumptions sub-level sets of f.
3. In section theorems. Furthermore, we explore the possibility of using 3 to craft a reconciliation strategy beating the naive ones (always reconcile, never reconcile), and shows that these strategies lead to RMSE reduction.
empirically assessed their correctness. The results confirm the soundness of the
To case the application of these thoorems and roconciliation methods, we release JAX-based python library, JNLR. https: /ggggtttth.oooo comb hstpssssssssae dacd-isae isaac/JNLL.
2 Nonlinear reconciliation problem
We would like to generate predictions for n signals, 2 E R which are known to satisfy one or more nonlinear
constraints encoded by the level set f(z) = 0 a vector-valued function f R" + R with m continuous constraints (1 < m S n). The nonlinear reconciliation problem minimally displacing the original forecasts è onto the constraint manifold f(z) 0 can be written as:
8 = 8(2) == arg min || - 2|[w, SE
(1)
where we assume M {a E Rd Under this assumption, M is an m-dimensional differentiable manifold, with codimension m, embedded in R" Here the dimension of M refers to its intrinsic manifold dimension, which could differ from the ambient dimension in which it is embedded. We define the matrix norm as ||æ||w = zTWz and W > 0 is a weight, matrix assumed to be symmetric and positive definite (SPD). SPD) assumption ensure a strictly positive objective. The mapping 8 IR" + M is called the nonlinear reconciliation operator. The linear reconciliation problem subsct of is usually referred to as bottom time series. The hierarchical reconciliation problem further subset, in which A encodes (usually unweighted) tree structure, Oij {0, 1).
[()) 0} where / Rd Rm is smooth and 0 is a regular value of
[ where M := (2 E R" 2 As, 0}. E (n-m)) and 26 E R"-m
The nonlinear optimisation problem in equation [ can be solved by forming the Lagrangian
C(2, (2 (=-2) 2) + IT=),,
AER".
A stationary point is characterised by V.C= = and V,L 0. Collecting these conditions gives the vector equation
V 2W (2
F(2,A)==
J= Vf(z) mm
(3)
V
f(z)
Starting from (2k, A1), Nowton-Raphson step (6z, 5X) solves
2W
J
8z
-F(zk Ak),
(4)
82
Jr Jr(zk,
after which update
(0411:6411 (Ek (82,
(5)
3

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Existence, uniqueness and convergence Since M = E R" [f(z) = =0} is closed set due to being continuous, and since the objective of equation 1 is coercive, minimum for equation | Jawaas exists. Since the objective is convex, the solution is unique only if M is convex. For the case of non- manifolds, under standard assumptions, namely i) f C2, ii) the constraint Jacobian is full row rank: rank(J(=x)) = m, iii the reduced Hessian of the Lagrangian positive definite on the tangent space at 2, the Jacobian JF (2, is non-singular and the Newton iteration is locally quadratically convergent for initial guesses sufficiently
close to (Σ, where (z, A) is a solution of Problem 1 Nocedal & Wright, 2006 Dennis & Schnabel 1983
Ortega & Rheinboldt 1970
In practice, for non-convex function, a globalization technique is usually used. In appendix A.5 we pro- vide the pseudo-code of a solver using line search, which is also present in the released python package. This algorithm instantiates a classical augmented Lagrangian framework for equality-constrraine constrained nonlinear optimization, as described e.g. by and [ertsekass] (2016) and Nocodal Wright (2006). Under the same assumptions previously stated for local convergence of the Newton- Raaitonn-- solver and standard regularity assumptions of the augmented Lagrangian framework ((existence of KKT point, Lipschitz continuity of the derivatives, and sufficiently accurate solution of the subproblems), augmented Lagrangian methods with lino-scarch-base based quasi- Newton inner solvers are known to be globally convergent in the sense that any ac- cumulation point of the outer iterates satisfies the KKT conditions of the original constrained problem. We do not derive new convergence results here, but we rely on this well-established theory; in our experiments, the algorithm converged robustly for all tested forecasts.
3 Error reduction theorems
In the case of linear constraints, has been shown that reconciliation via orthogonal projection always reduces the overall point-wise prediction error. This result relies on the Pythagorean theorem and was demonstrated in Panagiotelis et al. (2021). In this section, we derive analogous theorems for nonlinear constraints.
All proofs are based on a geometric interpretation of problem I as illustrated in Figure ! which depicts the reconciliation process for two examples of predicted points, denoted by é.
Let be prediction of the true point 2, and let denote the orthogonal projection of & onto the manifold M. We define three key vectors: the original prediction displacement after projection = - 2, and the reconciliation adjustment 8, E 2.
& the reconciled displacement
The condition for RMSE reduction can then be stated as:
|| 2 ||01|
(6)
Using the vectorial relation å = 6g + 5 this becomes:
||,_|| 1 + 26 8 2 0
(7)
that can be rewritten as:
| 11
5
(8)
3.1 Constant sign curvature hypersurfaces
A hypersurface in R" is smooth submanifold with codimension 1 that can be locally defined as the level set
M = {a E R" f(=)= c), where f n IR: is smooth function and is regular value (i.e., v (2) * 0 for all 2 E M). As an example, Figure 1 shows the hypersurface given by M = fz E IR: f(z) 2 y x² = 0} as thick black line, while blue and violet represent regions of the negative and positive values of f(z).
For hypersurfaces with a constant sign curvature, we can exploit the fact that they possess a supporting hyperplane at the point of projection. If the forocastod point 2 is on the right side of the hypersurface, it will see the surface bending away from it,, which sufficient to guarantee reduction of the error independently from where the true point is located.
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y= TyM
3.24
2.43
RMSE red.
1.62
0.81
0.00
6
-0.81
II
-1.62
6.
-2.43
-4.05
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
Figure 1: Conceptual plot for Theorem (2 with negative f(2) in the bottom right) and Theorem 3 with positive (3),, shown along atoms of its predictive distribution and their projection onto M).
Theorem 1: Constant sign curvature hypersurfaces
Let M = (=: 0} C R" be hypersuurace defined by J Rn R E C with Vf(=) + on M and with conver sub or super level sets. Given prediction E R" denote by & E M its orthogonal projection onto M. A sufficient local condition, that can be computed from the value of at the forecasted point & and the Hessian at the projection point &, for the reduction of the RMSE after projecting & inlo the following:
sign(f(2)) Amin (Han (2)) >0
(9)
where Htan denotes the Hessian H = fz) restricted the tangent space of M (see appendia A.2 and Amin denoles ils smallest eigenvalue.
Proof 1 A bounding condition for equation to be positive is 6 2 O J. That is, if we can show that this condition holds for arbitrary values of the theorem holds. We proceed this way since depends on where the true point will be on the manifold, which is unknown at prediction time, the need of bounding the expression for ang value of 8. Since 8x is orthogonal projection onto M, it is parallel to the gradient al Σ, so can rewrile.
1N5(2) 2 0 Võ
(10)
The sign of p is given by the side 2 is located with respect to the level set f(z) = 0, that
sign(u) sigg(((E))
(11)
since if f(2) is positive, the projecting vector 8x will oppose the gradient and vice versa.
Since M is the boundary of conver sub or super level set of f(z), it has supporting hyperplane in ž, that implies the sign of ( constant on M w.r.t. J. This also implies:
sign (()) ssign()nss (Hran(3) (E)))
(12)
(see A.3 for the proof). Putting together equation condition Theorem
equation 11 and equation
12 we obtain the sufficient

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Corollary 1: Constant sign curvature hypersurfaces
Under the some ossumptions of Theorem 1 defining o PV f(),, orthogonally projecting 2 into will decrease RMSE if the following condition holds:
<0 Vf f(=) < 0} conver
(13)
if fz f(z)(( 0} conver
Proof of corollary 1
The projection vector 8n at ā is by definition parallel to ff()) so that 6z WV
holds. Since sub-level set of f(z) is convee iff Amin (Hrn)) >> 0 (see A.2 considering equation 17 we can re-write condition
g as equation 13
Corollary | states that, knowing the sub or super level sets f() are convex, we can restate Theorem using the gradient at without computing the Hessian. We will use this fact to craft a theorem for vector valued functions in the next section.
3.2 Multiple constraints
The next result extends Corollary 1 to manifolds defined by multiple constraints, veclor-valued functions. The key idea is the same: if each defining function fi has convex sub or super level set, and if the projection direction lies in the positive cone spanned by the gradients at projection point [VA(E)] then this direction defines a supporting hyperplane to the intersection C = n C, of the corresponding convex sets C, (z fe(2) < 0}. This yields a uniform decrease of squared error. Figure illustrates the ruction.
25
M
z
Vf
Vi>
y
Figure 2: Concept for Theorem 2 with f (fi, f2) --- x² =]. The zero level sets fi 0 and 0
are shown in blue and violet. The intersection C = {fi 2 < 0} is in green. If the projection direction
is a positive combination of the normals (²), V f2(),, it defines a supporting hyperplane for C (gray).
Theorem Manifolds of codimension
with convex/concave level gcometry
and assume V(((z) + Let M = f= E Rd [(z) 0} with f = (fi,..., R -R" C
and rank (z) = m on M, where ((z) ff()). Suppose that for each i, the sublevel set C, = { fi(z) s 0} is convex. For any & E R= let M be orthogonal projection and define
8= 2
8(2) :=)==---

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If there exists I& such that
0x = Σ ())
(14)
im1
then for every 2 E C = t Ci > - 2| 2 || - 2|2; in particular, the inequality holds for all 2 E M.
If instead each fo is concave (equivalently, the superlevel sets {fr 0} are convez), the same conclu- sion holds with the sign in equation 14 reversed: 8n Ei pi V.())..
Since C₁ is convex and f((2) O, the hyperplane with normal Proof 2 Fix 2 and let C n C. Vf((z) supports Ci at E:
fi(z)' - 5) & 0 for all E Ci,
hence also for all z E C. If equalion
14 holds with pa 0, then
6 8(2)
M4 (2 -2) 0.
1=1
Using = 8n 0(2),
11]=~10(==)))))))))))))))))))) +2 - > |
which proves the claim. In the concave case, the supporling inequality reverses sign for the conver superlevel sets, yielding the stated flip of equation
3.3 Applicability of Theorem 1 and
Theorems ! and 2 provide sufficient conditions for a decrease of RMSE, conditioned on whether the forecast lies in the relevant sublevel or superlevel region of the function(s) defining M. How often those conditions hold depends on the geometry of M.
We estimaat m > 1, we sample points 2; E M. For each we draw an isotropic error NN(, Σ) with Σ = oj In at increasing noise standard deviations σ₁, form the forecast & 2 + E and compute its projection & by solving equation] We then: (i) check the corresponding theorem's condition at & and record the indicator
priori likelihood as follows. For several hypersurfaces (m 1) and manifolds of codimension
of it holding, and (ii) record whether reconciliation reduces squared error, i.e. 1 ž 1| < || - ).
Averaging these indicators over yields the empirical probabilities shown Figure 3 (left: theorem's condition
holds; right: squared-error reduction).
Across the tested manifolds, the sufficient conditions identify only subset of cases that actually benefit from reconciliation. Moreover, real forecasting errors are rarely isotropic around nonlinear M. The next example shows that even with optimal mean squared prediction, forecasts can systematically fall into the region where Thcorems and 2 do not trigger.
Example 1: Optimal prediction on parabola
Let M {2: f(2) = =0 be parabola, with f(2) 2 R + R where = (21, (21,22) E R?. Given dalasel D = [(21,1, ) the independent leasl-squares predictors are the empirical means
x 1/2 Σ
21,
22 = 1 Σ = 1 Σ
22, 2
1
21,5
=
2+2
where the inequality is Jensen's. Hence f(2*) 2 23 < i.c., the optimal independent prediction lies in the sublevel vegion (22 2 2?}
Therefore, the rates in Figure 3 are optimistic regarding the frequency with which the sufficient conditions will hold in practice: in many realistic settings, is likely to lie in the region where reduction in error

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is not guaranteed by Theorems andd 2
This motivates the probabilistic guarantees for general nonconvex
manifolds of codimension m developed in the next section.
Th 1 and 2
Actual P(RMSEreduction)
1.0
1.0
Manifold
manifold
vv. ww..uadd quad
0.8
0.8
4d-4d
Probabiliyy
p_lii-ouad
0.6
0.6
hs, paraboloid
mixed quadratic
hs_ js_exponential
0.4
0.4
hs, quartic
manifold type
vector valued
0.2
0.2
hypersurface
0.0
0.0
10-2
10
102
10-2
10
102
Noise Level
Noise Level
Figure 3: Left: probability observing a positive condition from Theorem 1 andd] under isotropic Gaussian perturbation of points on the manifold. Right: empirical probability of reducing RMSE by reconciling. The axis shows the noise level ds.
3.4 Non-constant sign curvature manifolds and probabilistic estimators
In Theorem used the orientation condition in equation Logether with equation 12 sufficient error-redde reduurionn result for constant-sign curvature hypersurfaces. However, shown in figure these conditions are consorvative: if the prediction & lies on the side where M bends towards 2, the Theorem can't 't predict if the RMSE is going to decrease. For those cases, and for general manifolds without constant- sign curvature, strictly deterministic guarantees are not possible without knowing the true location of z.
to obtain
However, we shift the perspective from deterministic guarantees to probabilistic ones, we can still leverage the general inequality described in equation [ which says that the post-projection error vector å may "climb" along the projection direction & by at most half its length. Geometrically, this defines half-space of candidate true points for which reconciliation reduces the RMSE. Figure Figure@illustrates iilustrates, in two dimensions, the corresponding half-plane for the left. prediction point. For surfaces that can be sampled, this half-space intersects M in a region where we will observe a reduction.
While informative, the geometric condition in equation reduction. We propose a practical probabilistic estimator to answer: What the probabilily that projecting the prediction onto M reduces RMSE?. Since the only unknown in equation 3 is the post-projection error 8, which is fully determined by and 2, this question reduces to estimating the probability that the truc point lies in the favorable half-space. The following discussion is general in terms of reconciliation method; in particular we don't assume an orthogonal projection for the reconciliation.
S docs not directly quantify the likelihood of error
3.4.1 Coherent probability distributions nonlinear manifolds
We recall the definition of reconciled probability distribution introduced in Panagiotelis al. (2023) and Zambon et al. (2023). This concept underpins the probabilistic guarantees derive for manifolds with non-constant-sign curvature.
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Definition 1: Coherent probability distribution nonlinear manifold
Let D E P(R") be predictive distribution on (R" Ba-) and let 8 R M be mores-leasurable reconciliation operator whose image equals the constraint manifold M == (2 IR" f(=) = 0}. The
coherent predictive distribution ù &E P(M) is the pushforward
ÿ = S
Fe FEBM.
where BM = {F C M Bpn)
3.4.2 Probability of RMSE reduction
Let h IR > {0, If be the Heaviside function h(u) 1 [a>00-- For cach time t, let Ze be random vector with (unknown) true and predictive distribution V4, Dr on the ambient space, and let & denote the (deterministic) reconciled point at time t. We assume vt, dt absolutely continuous w.r.t. reference measure À such that their probability density functions exist, dur == fu. dis fo.. We write d == ãt 2 for the reconciliation adjustment. Define the scalar map:
dr(=) == oT,(== 2)) + =10
(15)
From condition equation projection from & to &n reduces the RMSE at time / if and only if d). > 0. Thus the true probability of RMSE reduction at time t is:
6 = 12 ( = f ( | h(yy) dpe(y) = (Y)),
(16)
where 14 = o 2/ is the distribution of the scalar random variable 4; = Φe(Ze) on R, and Y = h((,)) is
the binary indicator reduction.
In practice, the true law 14 is never observed, and we just have access ex-post to a sample drawn from it time t, 2 v4- We therefore approximate by consistent predictive distribution in from which we To ease geometric inluition, Figure 4
retrieve an estimated probability of improvement & == Ex, depicts the relationship between vt, bt, pt, Pt, de and the estimators ez, ēz
3.4.3 Monte Carlo Estimator
Given samples (atoms) &,, ~ in with normalized weights ti.s (where #e,s = 1/S for unweighted Monte Carlo) we define the estimator:
1 s
Emc, =
Σ (()((E)))).
(17)
S ==1
Without loss of generality, and to ease the notation, we will just consider the unweighted Monte Carlo.
Lemma 1: Consistency of the Estimator
Let ih be predictive distribution at time and let 2,, E.s be i.i.d. draws from Vc. Then ēme, (with equal weights) is an unbiased and strongly consistent estimator of ea under the predictive law it i.e., Blême,,) = and Eme, a.s. Ee as S oc.
By construction, Y, are i.d. with E[Y.] Ee thus the result follows immediately from the Strong Law of Large Numbers.
One possible practical procedure to obtain ê,c, is the following:
1. Obtain the reconciled point forecast. via the nonlinear projection operator: E, 8(2))

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2t
& (++)
f H4
S (2),a)
2t.s
0.6
Sir
zt
1=
fire
h (60)
tt
fūr
Figure 4: Conceptual diagram for the probabilistic RMSE reduction. The unknown true distribution 7 and the predictive distribution i, mapped by 6e to the scalar probability density functions of 14 #1, and tr Φ1 2/- Applying h yiclds the truc and predicted probabilities of RMSE reduction = E (+)) and é, = E,,,, [h( respectively.
the depicted realization, 60% of fu, is below the dashed line defined by (x,2,) 0, so the true RMSE roduction probability is e 0.6. However, the predictive distribution entirely lies below the dashed line, resulting in an estimated probability
1. The observed žt, lies above the dashed line, (z)) < 0 and the projection to & would increase the RMSE at time (and thus = 0).
2. Starting from atoms of non-reconciled predictive distribution ât.,, obtain the consistent samples alwa always via projection ž,, {8( By definition | this defines sample-based representation of a coherent probability distribution on M.
3. Count how many reconciled samples fall in the right half-space via equation 17 using ~..- = ot, E.
We emphasize that the estimator equation 17 does not require the use of an orthogonal projection, contrast to Theorem | and Theorem 2 This represents a distinct advantage, since, least in the case of linear constraints, non-orthogonal projections have been shown to yield even greater RMSE reductions in many settings.
3.4.4 Calibration Bounds
While Lemma guaarantees convergence to the predicted probability é,, it does not guarantee that matches the true probability ez. A more interesting result would be to bound the distance of the Monte Carlo estimator Emot from the realized outcome Y,, |ême,t - Y,l, or the distance of the Monte Carlo estimator from the true probability of RMSE reduction, leme,t - Gel. The distance @mme, - Ytl cannot be bounded tightly for a single instance t because we only observe a single binary outcome Y = hl and never have accoss to the true distribution it (see appendix A.4). However, we can derive rigorous bounds on the calibration of the estimator êmc, over an archive of forecasts.
Theorem 3: Finite- Conditional Calibration Bounds
Let us define the true conditional probability of RMSE reduction at prediction level as:
:= = E -= =e),
(18)
For any E (0. 1), we define the binned archive
(e ((e,, u)) fl...., N), E fe 0 A]).
(19)
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[A]e)] and number of successes ka(e) == EiEAs (e) Yú, with yi {0,1) binary indicator for the successful reconciliation and E, enror reduction probability under the forecast law.
with bin size (e(e)=
I- Bounds for Te): Assume each bin (e) contains i.i.d. scenario pairs. Then, with probability at least 1 1-B over the random mechanism producing A (c), the true conditional probability is bounded by
(3, Na(e), ka(e)) S (() S ea(()Na(e),,k((e))
(20)
where [et, ev] denotes the Clopper Pearson confidence interval for binomial proportion with (a(e) trials and (e) observed successes ot confidence level 1 1.8. 3.
II Instantaneous bounds for ler ême,l: Assume the conditional distribution is stationary such that = (e) Consequently, for a deployment time where equation 17 predicts ême, the instanlaneous eslimation error is hounded by:
Lerr S Slee- S Uerrs
(21)
where the magnitude bounds are determined by the geometry of the interval relative to the estimate
e:
Uen == max([eu les ee);
(22)
E1 -
if e1 > e,
Ler
ev iee <e,
(23)
otherwise (if e E (el.
Proof 3 I Bounds for a(e) We condition on the rundom event E = e; for all E A (e) and, wnder the i.i.d. assumption of the archive, the outcomes ys are independent Bernoulli trials with success probability x(e) defined in equation 18. Therefore, the count of successes follows a Binomial distribution: K ~ Bin(k: =())). The Clopper Pearson interval [eI, is constructed by inverting the Binomial hypothesis test. By defirition [Clopper Pearson 1934), it satisfies:
Pk~Bin(k,M(O) (M(e) (e)( 21-6.
This establishes that the random interval eul covers the true conditional probabilily T(e) with probability at least. 5.
II Instantaneous bounds for Em,rl Under the stationarity assumption, the true probability time e, =(e). We condition on the event that the interval covers the truth, i.e., eL S e, S eu, which holds with probability 1 8. The estimation error is lez ēme,t- Since ēme, is fired and e is constrained to the interval [er, eul, the ervor is constrained by the minimum and mazimuun distances from the scalar to the set eul.
The mazimum distance Uerr occurs at one of the endpoints, hence Uar = maxpelez,eul lp el max(leL - el,leu el).
The minimum distance Ler is 0 if lies inside the interval. If the interval is entirely above (er > e), the minêmum distance is e. If the interval is entirely below (eu < <e,,, the mindmum dislance is E - euu-
Combining these cases yields the definitions in equation 2 and equation
Use and data collection: Informative reconciliation guarantees arise when the interval [ez, eu] is narrow and statistical uncertainty low. For example, one-sided lower bound IP(m(e) 2 ex.) with en 2 0.9 provides
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strong evidence in favor of reconciliation. Similarly, an upper bound P(x(e) ey] with ey I 0.1 indicates that reconciliation is unlikely to be beneficial. Both cases are informative from statistical standpoint. In contrast, an interval [ez,ev] = (0.1,0.9) offers a weak statistical support and does not convey a clear expoctation of positive error reduction. On the other hand, if the interval is narrow but close to 0.5, e.g., [eL.ec] = [0.45, 0.55]; the statistical (epistemic) uncertainly in the estimate low. Still, the results of the previous thcorems do not provide a definitive indication of whether reconciliation should be applied in this case. This is due to the geometry of the problem (manifold) and natural variability in the probabilistic predictions and data. Aiming for reliable reconciliation with high confidence, that is, high ez and low 8, archive A (e) with minimum size N is required. The data set size may be computed numerically by inverting Theorem 3 however, this requires specifying a target & (or expected rate k/N), which is not always controllable. Yet. inversion offers a simple and practical guideline for experimental design and selecting a sufficienlly large sample size for a target e. In fact, the minimal required sample size N may be increased by further exploring the generating process and cnlarging the archive, i.c., by collecting additional scenario pairs (ei, ys) verify whether reconciliation elfectively reduces the error. This approach can help mitigate epistemic uncertainty by providing a more representative coverage of the forecast law at the target prediction level However, practical limitation arises when the forecaster, data, reconciliation, and manifold jointly lead to limited number of hits within a specific bin E [e A, e + A]. In such cases, even increasing the overall archive may not sufficiently decrease uncertainty for that bin, as the number of realizations directly constraining Hee) is inherently limited. Consequently, while enlarging N a viable strategy to strengthen statistical guarantees, the achievable reduction in epistemic uncertainty is ultimately constrained by the intrinsic distribution of forecasted errors and the underlying geometry of the forecaster and the manifold. This underscores the importance of both careful experimental design and awarencss of the process-specific limitations when interpreting informative reconciliation guarantees.
4 Numerical tests
To facilitate the application of the proposed theorems, we provide an open-source JAX-based Python library, http://gthbbb.oooomsssslaaaaaaaa /scpp/siiaacdisssccc/aaaaaeaeaae JNL.R which supports just-in-time compilation and GPU acceleration. h
We use this library to conduct numerical tests for the presented theorems. The tests cover a collection of manifolds with codimension and 2, including cases with constant-sign curvature and convex sub-level sets. The applicability of each theorem to the different classes of manifolds is summarized in Table
Table 1: Applicability of the presented theorems.
constant sign curvature non-constant sign curvature
codinension 1 codimension >1
2
3
Furthermore, as stated before, equation 17 also applics when using non-orthogonal projections.
4.1 Experiments setup and evaluation metrics
The goal of the numerical tests is to evaluate the presented theorems in a realistic forecasting scenario. Reconciliation is usually used as post-processing technique, after each of the n variables of the multidimen- sional point has been predicted independently by a statistical or machine- are independently generated, they won't perfectly lic on the manifold M defined by the constraints. Recon- ciliation corrects this by projecting an unconstrained forecast onto M, enforcing coherence. The experiments in this section simulate a typical forecasting pipeline. We start by generating data from a autoregressive process. This allows us to control the variance and correlation of the different components of and change them systenatically. This is a standard setting in simulation studies for forecast applications. Secondly, we independently fit standard regression models (TightGBMM), which are then used to obtain our unreconciled
model. Since predictions
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forecasts 2. Those are then projected via 8, which allows to investigate the validity of the RMSE reduction theorems. A more technical description of the setup follows.
All the tested manifolds are defined by one or two graphs R2 R, so that M : {z E R² f(=) 0},
f = g(2,, 22) for manifolds with codimension and f= (21 22),22())22 92(21, 22)) ) for manifolds with codimension
2. This allows us to sample the manifolds by defining a two-dimensional data generation process spanning the surfaces:
21 01 w1
(24)
2+1 2
+ w2
where w1, W2 N(0,0²) are two independent random variables with normal distribution and 01, 02 [0, 1] are two arbitrary parameters defining the autoregressive process. We then use the graphs g to obtain coherent tuples, zt- We built datasel of T 10a samples D {=) applied a random reshuffle and trained LightGBM regressor to predict 2+41 from &; on 10% of D. At each time t, we obtained predictive distribution in terms of samples {(2r,))=m with S = 200 via bootstrap on a calibration set with 40% of the D, and we used the remaining 50% to Lest the theorems. Figure 5 shows a hypersurface (paraboloid) and a manifold with codimension >1 (quadraiic-exponentia) as examples of M. On both panels, three examples of predictions in terms of deterministic points ε, associated bootstrapped samples, their projection onto M, and true points are shown.
2
0.7
2
2
0.6
0.5
3.5
0.4
Z
0.3
0.2
2.5
0.1
2.0
0.0
Z
1.5
001
0.6
1.0
0.4
-0066
-00.4
0.2
-02
-0.4
-0.2
-02
0.0
0.2 X
0.0
0.0
0.2
0.4
D.5
0.6
-0.4
-0.6 -0.4
-.2
0.0
0.2
0.6
0.6
0.4
Figure 5: Examples of test cases in terms of tuples of predictions (green dots), ground truth (red triangles). and samples from predictive and reconciled distributions (gray and blue dots). Left: paraboloid case, Right: vector-valued function. In this case, just one tuple is plotted, but two points are visible since : + R.2
For Theorems and we retrieved the number of predicted RMSE reduction cases and the false positives (cases in which the theorems predict decrease of RMSE while observing an increase). We anticipate that we observed total of 0 false positives across all the functions.
Since equation 17 provides a probabilistic estimation, we assess it with a randomized study. We generate 25 studies with associated datasets D, of 10 points from M using the data generation process equation 24 with random parameters 0, and lz. We follow the previously described procedure to obtain predictive distributions via bootstrap on each test set. For each point on the test set we then evaluate I7 ême,t- To estimate the probability of RMSE reduction defined in equation 18 adopted binned calibration
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estimator. We used fixed width of A 0.01 and estimated ((e)) as
1 N
$ (ex)
y
(25)
(EA (mL)
where AX (ex) is the binned archive to which e, belongs. As before, is the binary outcome for the ex-post condition (i.c., whether reconciliation led to lower RMSE after the uncertainty resolved).
Since we equip the calibration estimation with upper and lower boundaries via Clopper-Pearson intervals, we also test for the binary coverage of the intervals, defined as:
CL. 0.5 if ||0|| < [|8]
c=
E q C
(26)
EUJ < 0.5 otherwise
iEDio
where eu, e, are the upper and lower bounds defined in equation 20
Finally, considering the case in which the forecasting lask doesn't explicitly require E M, we investigate if can be used to craft an effective reconciliation strategy, reconciling just the points for which the estimated probability of a RMSE reduction higher than a given threshold 0:
St = s(ir)
Emo,t >0 otherwisc
0 =
(27)
21
The results are shown as the reduction in RMSE from the baseline RMSE [|ô]] (never reconcile), normalized for the RMSE of the optimal strategy, where the forecast reconciled if this guarantees a reduction in RMSE:
s(2,) if 22
(88)
otherwise
That is, 2f represents the best possible ex ex-post strategy, or equivalently, the optimal reconciliation strategy with perfect knowledge of the future. The final normalized score is:
|0
12 2z
RASE rel,opt
(29)
|0
-| 2x
4.2 Hypersurfaces
Plots the tested hypersurfaces are shown in 12 in the appendix. Table shows the ratio of cases with an observed RMSE reduction (first column), with a predicted reduction from Theorem (sccond column), and of false positives, for different levels noise affecting the data generation process described in equation [24] Across all tested hypersurfaces and noise levels, we observed zero false positives, providing strong empirical support for Theorem !
In Figure i two examples of calibration plots are shown for constant sign curvalure (paraboloid, left) and non con-constant sign curvature (Himmelblau, right). Blue points show the collected 1.25 x 10t (25 studies on the 50% test sets of the 10a datasets) RMSE differences | ||ō|| before and after reconciliation. Lines show the moving averages ((e) from equation 25 The top and bottom panels refer to the upper and lower bounds obtained from Clopper- Slopper-Pearson Pearson intervals, respectively, while the middle one shows f(e) for the nominal response of equation 17 higher inter-study variance is expected the contral region of the plot.
Table reports the binary coverage from 26 We see that the coverage tends to decrease with the noise a, possibly due the deterioration of the forecasts. the predicted distribution not representative of the true distribution v, expect the test to be unreliable.
Figure 7 shows boxplots in terms of A rel.opt defined in equation 29 Each boxplot contains A RMSE from the 25 randomized studies. From the definition of A A rel.opt positive values indicate a reduction in RMSE compared to the unreconciled case, while score of 1 represents the best binary reconciliation strategy.
rel.opt
RMSF
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||
||8]
Th. reduction
Th. 2 FP
0.1
Abs
0.59
0.13
0.00
Exponential
0.87
0.46
0.00
Mixed
0.37
0.15
0.00
Paraboloid
0.34
0.10
0.00
Quartic
0.34
0.11
0.00
= 0.3
Abs
0.59
0.13
0.00
Exponential
0.68
0.36
0.00
MixedQuadratic
0.38
0.15
0.00
Paraboloid
0.33
0.10
0.00
Quartic
0.33
0.11
0.00
0.5
Abs
0.59
0.13
0.00
Exponential
0.60
0.36
0.00
MixedQuadratia
0.38
0.15
0.00
Paraboloid
0.33
0.10
0.00
Quartic
0.30
0.12
0.00
0.7
Abs
0.59
0.13
0.00
Exponcntial
0.55
0.33
0.00
Mixed Quadraticc
0.38
0.15
0.00
Paraboloid
0.33
0.10
0.00
Quartic
0.29
0.09
0.00
= 0.9
Abs
0.59
0.13
0.00
Exponential
0.52
0.33
0.00
MixedQuadratic
0.39
0.15
0.00
Paraboloid
0.33
0.10
0.00
Quartic
0.28
0.09
0.00
Table 2: Predictions and false positives for Theorem for different hypersurfaces with constant sign curva- ture, increasing levels of noise a.
0.1
a = 0.3
0.5
0 == 0.7
= 0.9
Rosenbrock
0.59
0.52
0.49
0.47
0.45
Ackley
0.59
0.50
0.37
0.28
0.21
Rastrigin
0.63
0.51
0.40
0.31
0.25
Quartic
0.65
0.64
0.65
0.65
0.66
Paraboloid
0.67
0.67
0.67
0.66
0.66
MixedQuadratic
0.69
0.68
0.67
0.66
0.66
Himmelblau
0.75
0.68
0.63
0.57
0.54
Abs
0.81
0.81
0.81
0.81
0.81
Exponential
0.84
0.68
0.57
0.57
0.58
Table 3: Binary coverage from equation
26 for different hypersurfaces, increasing levels of noise
The first boxplot represents the strategy in which the points are always reconciled, while the others represent the stralegy of equation 27 with increasing values of 0. For the exponential manifold, the best strategy, close to the optimal, is always reconciling. For the Rastring manifold, reconciling always worsens the RMSE, and choosing 0 0.6 only slightly improves the RMSE over the unreconciled case. For the other tested hypersurfaces, reconciling improves the RMSE, and the best reconciliation strategy based on equation uses threshold of 0 0.5 0 = 0.6.
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0.6
1.00
0.4
0.75
0.75
0.50
0.50
0.0
0.25
-1
0.25
0.00
0.00
0.0
0.2
0.4
1.0
1.00
1.00
0.4
0.75
160 10
1-
0.75
0.50
0.50
0.0
0.25
0.25
0.00
0.00
0.2
0.4
0.8
1.00
0.75
jól 16
1-
0.7
0.50
0.25
0.00
0.0
0.2
0.4
0.6
0.8
0.0
0.2
0.4
0.5
0.8
1.0
(a) Panel A: Paraboloid
(b) Panel B: Himmelblau
Figure 6: Calibration plots for equation 17 for two examples of constant sign curvature (left) and non- constant sign curvature (right) hypersurfaces, for noise level a = 0.3.
4.3 Manifolds with codimension 1
We repeat the same tests for manifolds with codimension>1, using Theorem 2 for manifolds with convex sub-level sets. Figure 13 in the appendix shows plots for the tested manifolds. Table shows the ratio of cases with an observed RMSE reduction (first column), with predicted reduction from Theorem (second column), and of false positives for manifolds with convex sub-level sets. Also in this case, no false positives are present. This empirical finding is fully consistent with Theorem 2 prediction that false positives should never occur under its assumptions.
Figure g shows the calibration plots for a quadratic-quartic and for a bowl-sin manifold. Also this case, for the manifold shown in the first panel, we can observe some underestimation the probability of an improvement due reconciliation in low-density zones of the plot (low probabilitics). As before, this is likely due to the forecast being unreliable these conditions.
The binary coverage is shown in Table 5 where the coverage is lower under high noise for the ring-trig and bowl-sin manifolds, which are the most irregular ones. Also in this case, this is likely the effect of the forecast i becoming unreliable.
Finally, Figure 9 shows boxplots in terms A RMSE for different manifolds. Also this case, for the rel.opt
saddle-poly, exp-cosh, and Rosenbrock style manifolds, always reconciling the sample close the optimal strategy. For the other manifolds, for the hypersurface tests, the best reconciliation strategy based on equation 17 uses throshold of 0.5 or 0.6.
5 Note geodesics and optimal deterministic predictions
This paper presented RMSE reduction conditions for nonlinear manifolds. However, for some forccasting tasks, other metrics could be more relevant. we have coherent forecast distribution & supported on M, finding the best point forecast is not as forward in the linear case. In general, given sample-based
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LAMSE
0.5
ARMSE
0.5
Aral opt
0.5
0.0
Paraboloid
Mixed
Exponential
0.0
0.5
00.5
ARMSE ope
rell.
-10
-1.0
0.0
-15
Quartic
Rastrigin
-1.5
Rosenbrock
20
2.0
0.5
0.5
0.5
ARMSE
AAAEE
0.0
0.0
Abs
Himmelblau
-05
Ackley
always
m
always
alwa
Figure 7: Boxplots of normalized improvements in RMSE for different reconciliation strategies for hypersur- faces.
representation of 2, ( the best deterministic point can be found solving:
arg min
(,, 2,)²
(30)
In the linear case M R" and under RMSE loss I, the minimizer the weighted average of the samples, while when M nonlinear manifold, this must be solvod numerically. For general Manifolds, problem 30 is also known Fréchet mean (Bačák) 2014 Lou et al. 2020), and in general non-trivial to solve. For forecasting tasks, it may be more meaningful to consider the geodesic distance the loss rather than the RMSE. Defined b) > M a curve on the manifold, the geodesic between and b is the distance of the shortest path on M between these two points:
g(a,b) inf [ [1)(
(31)
where the Riemannian metric p= (Pa)zEM is smooth collection of inner products Pa T,M T.M - R. The following example ralustrates the importance of considering geodesic distance for forecasting tasks.
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||
||ô]]
Th. reduction
Th. 2 FP
0.1
4d-4d
0.36
0.09
0.00
bowl-sin
0.33
0.10
0.00
e2-s2
0.34
0.10
0.00
quad-qq
0.39
0.04
0.00
quad-quad
0.33
0.10
0.00
= 0.3
4d-4d
0.35
0.09
0.00
bowl-sin
0.36
0.07
0.00
c2-s2
0.32
0.09
0.00
quad-qq
0.39
0.02
0.00
quad-quad
0.33
0.10
0.00
0.5
4d-4d
0.32
0.09
0.00
e2-s2
0.29
0.08
0.00
quad-qq
0.42
0.01
0.00
quad-quad
0.33
0.10
0.00
0
0.7
4d-4d
0.27
0.09
0.00
e2-s2
0.28
0.07
0.00
quad-qq
0.48
0.01
0.00
quad-quad
0.33
0.10
0.00
0.9
4d-4d
0.25
0.09
0.00
c2-s2
0.27
0.06
0.00
quad-qq
0.52
0.00
0.00
quad-quad
0.33
0.10
0.00
Table 4: Predictions and false positives for Theorem for different M with codimension>> and convex sub-level sets, increasing levels of noise a.
σ 0.1
0.3
0.5
0
0.7
σ 0.9
quad-qq
0.61
0.64
0.62
0.61
0.59
ring-trig
0.61
0.23
0.11
0.10
0.07
e2-s2
0.64
0.66
0.69
0.71
0.71
4d-4d
0.64
0.66
0.70
0.71
0.72
exp-cosh
0.64
0.64
0.64
0.62
0.59
quad-quad
0.65
0.64
0.63
0.64
0.63
bowl-sin
0.65
0.46
0.37
0.30
0.25
saddlc-poly
0.69
0.70
0.70
0.67
0.65
rosenbrock
0.82
0.68
0.60
0.54
0.54
Table 5: Binary coverage for equation equation noise a.
17 for different M with codimension>>1 1, increasing levels of
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1.00
1.00
0.75
0.75
0.50
[6]
0.50
0.25
0.25
0.00
0.00
0.8
1.0
1.00
1.00
0.75
160 101
0.75
2-
0.50
0.50
0.25
0.25
0.00
0.00
0.4
0.8
0.8
161
2.-
jól
0.0
0.2
0.4
0.8
1.0
0.0
0.2
0.4
0.5
0.8
1.0
6V
(a) Panel A: Quadratic-quartie
(b) Panel B: Bowl-sin
Figure 8: Calibration plots for equation equation 17 for two examples of M with codimension > and convex (left) and non-convex (right) sub-level sets, for noise level a = 0.3.
Example 2: Optimal forccasts under geodesic distance
A heavy snowfoll forecust for tomorrow, with spotial probability distribution v(= = [x, y)), over the area surrounding large lake. The event is expected lo significantly disrupl road traffic. To reduce the economic impact, road maintenance crews be preemptively deployed to specific location along the road network, enabling a rapid response. Figure 10 illustrates two candidate deployment locations, žι and ス2, marked as green dols and selected by the decision maker. While ž2 is closer to the true disruption point (red triangle) in terms of Euclidean distance, deploying crews there would incur much higher operational costs than choosing ž1. This is due to the road network construints, as shown by the red and green geodesic paths, which represent the cost of relocating crews from each condidale point to along the road infrastructure.
6 Conclusions
This paper has advanced the theory of nonlinear forecast reconciliation through geometric lens, establishing both deterministic guarantees and a practical probabilistic framework. Our first key contribution lies in Theorems and which provide sufficient conditions for strict RMSE reduction when reconciling forecasts via orthogonal projection onto constraint manifolds M = {2 : f(2) = 0}. These deterministic results hold for codimension and higher when all component. functions f, exhibit convex sub-level or super-level sets.
However, our simulation study under isausoni errors revealed significant. practical limitations: these geometric conditions apply only to subsets of prediction points, particularly highly curved manifolds. This observation motivated our second major contribution- Theorem -hhichh addresses the fundamental question of when reconciliation is likely to help. By introducing consistent Monte Carlo estimator for Pr(1=- - 2|| Z ||= =]]), this probabilistic framework accommodates arbitrary manifolds without curvature assumptions or codimension restrictions, while providing confidence intervals.
Empirical validation confirmed the soundness of all theoretical contributions. Crucially, we observed zero false positives across all tests of Theorems and consistent. with their theoretical guarantees. The estimator
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1.0
0.5
0.5
ARMSE op
RMSE ral. out
0.5
ARMEE opt
0.0
0.0
0.0
quad-quar
quad pq
e2-s2
-0.5
1.0
0.50
AROSE
RASE
0.25
RMSSE opt
0.5
0.00
0.0
0.0
4d-4d
-0255
bowl-sin
Saddle-poly
1.0
0.0
1.0
ARMSE
0.5
RAASE Pail out
-05
RMSS felos
0.5
0.0
0.0
rosenbrock
-1.0
fing-trig
exp-cosh
always
always
Figure 9: Boxplots of normalized improvements in RMSE for different reconciliation strategies for manifolds of c.dimension>11.
Y1
Z
z
1'2
Z:2
Figure 10: Conceptual plot for the importance of using geodesics scoring rules in forecasting, Red triangle: true point z; green dots: examples of reconciled forecasts š,, ž2; blue lines: euclidean distances; red and green lines: shortest paths on M from predictions to 2.
from equation 17 demonstrated excellent calibration, with empirical coverage of confidence bands aligning closely with nominal levels. Most significantly, when using this probability estimate to decide whether to reconcile-rather than naïve always/never approaches-we we achieved mcasurable RMSE reduction across diverse manifolds.
Collectively, these results bridge geometric theory with forecasting practice: while Theorems | and 2 establish fundamental crro--reduction principles under idcalized conditions, equation [1 and Theorem 3 deliver a versatile decision-mall makinngaannnggn tool for real-world applications. This dual perspective enables more reliable forecast integration where theoretical guarantees meet practical uncertainty.
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6.1 Future research directions
This paper lays the foundations for fulure research directions in the field of nonlinear reconciliation. An interesting research direction is to craft analogous theorems using the geodesic distance instead of the RMSE. An interesting research topic is to find a function of the geodesic that is a proper scoring rule for some classes nonlinear manifolds. For example, the energy score proper scoring rule and it's a function of the Fuclidean distance. Another interesting research direction to find robust rules to set the threshold parameter 0 that maximizes the accuracy of the reconciliation strategy. Lastly, the RMSE reduction theorems and associated reconciliation stralegies could be applied to nonlinear state estimation problems.
Acknowledgments
This work has been funded by the Swiss State Secretariat for Education, Research and Innovation (SERI) under the Swiss contribution to the Horizon Europe projects DR-RISE (Horizon Europe, Grant Agreement No. 101104154) and REEFLEX (Horizon Europe, Grant Agreement No. 101096192). The authors would like to thank Lorenzo Zambon for his insightful feedback and valuable suggestions.
References
George Athanasopoulos, Roman A. Ahmod, and Rob Hyndman. Hicrarchical forecasts for Australian domestic tourism. International Journal of Forecusting, 25(1):146 166, January 2009. ISSN 0169-2070.
doi: 10.1016/j.jfoorosss.2000 www.so00.....0.07..... URL http://wwww.scencederetttttt http:///www..ccccceeeeee.....ooa ctt/////eceeeeeeearrccc/a////// 501692070080006911
George Athanasopoulos, Rob J. Hyndman, Nikolaos Kourentzes, and Fotios Petropoulos. Forccasting with temporal hierarchies. European Journal of Operational Research, 262(1):60-74, October 2017. ISSN 0377-
2217. doi: 2..1000//.jjor22117222... URT 50377221717301911
Miroslav Bačák. Computing Medians and Means in Hadamard Spaces. SIAM Journal on Optimization, 24 (8):1542-1566)) January 2014. ISSN 1052-6234. doi: 10. 10.1137/140953399 URL https://epubs.. siam. org/ Publisher: Society for Industrial and Applied Mathematics.
Souhaib Ben Taieb, Jiafan Yu, Mateus Barreto, and Ram Rajagopal. Regularization in Hierarchical Time Series Forccasting with Application to Electricity Smart Meter Data. Proceedings of the AAAT Conference on Artificcal Intelliignnn (1).), February 2017. ISSN 2374-3468, 2159-5399. doi: 10.1609 111100/aaai.33111.1111 URL http://.w...aaa...o.ooaaaaaaaaaa
Dimitri P. Bertsekas. Nonlinear Programming. Athena Scientific, 2016. URL
nonlinbon inbook. html
M. Brown Do Coutto Filho and J.C.S. De Souza. Forocasting-Aided State Estimation Part Panorama.
IEEE Transactions on Power Systems, 24(4):1667 1677, November 2009. ISSN 0885-8950, 1558-0679. doi: 10.1109 TTw.1////.....099922 2030099. URL http://ieeeeploree ieee. org /ttp///cmmmmtt/233355//////
Andrea Cini, Slobodan Lukovic, and Cesare Alippi. Cluster-based Aggregate Load Forecasting with Deep Neural Networks. In 2020 International Joint Conference on Neural Networks (IJCNN), pp. 1- 8, July 2020. doi: N0.1109/JCCNNSS0 200.0..2220.99000000 URL 9207503 ISSN: 2161-4407.
Andrea Cini, Danilo Mandic, and Cesare Alippi. Graph-based time series clustering for end-to-end hierar- chical forecasting. In Proceedings of the 4 1st International Conference on Machine Learning, volume 235 of ICML 24, pp. 8985 8999, Vienna, Austria, July 2024. JMLR.org.
C. J. Clopper and E. S. Pcarson. The Use Of Confidence Or Fiducial Limits Ilustraeed In The Case Of The Binomial. Biometrika, 26(4):404-133, December 1934. ISSN 0006-3444. doi: 10.1093/ 10.109//biomet// biomet/26.4..... URL Ctttp:://aai.crg//f..000////////
4.404.
21

---

doi: http://jssda.2111....3 Published in Transactions on Machine Learning Research (01/2026)
Giorgio Corani, Dario Azzimonti, João P. S. C. Augusto, and Marco Zaffalon. Probabilistic Reconeiliation of Hierarchical Forecast via. Bayes' Rule. In Frank Hutter, Kristian Kensting, Jefrey Lijffijt, and Isabel Valera (eds.), Machine Learning and Knowledge Discovery in Databases, Lecture Notes in Computer Science, pp. 211-226, Cham, 2021. Springer International Publishing. ISBN 978-3-030-67664-33 doi: 10.1007/978-3-0000 000-00000--7-- 13.
J.E. Dennis and Robert B. Schnabel. Numerical Methods for Unconstruined Oplimization and Nonlinear Equations/ / SIAM Publications Library. 1983. URL https: https://epubs.. siam. siiam.ogg/a//bbo//10.... 1137/1
9781611971200
Chiara Di Modica, Pierre Pinson, and Souhaib Ben Taieb. Online forecast reconciliation in wind power prediction. Electric Power Systems Research, 190:106637, January 2021. ISSN 03787796. doi: 10.1016/j. DnsT.2020.10660 URL https:///innkinghub elsevier http:///w./oooaaaa/////////////0
D. M. Dunn, W. H. Williams, and T. L. Dechaine. Aggregate versus Subaggregate Mod- els in Local Area Forecasting. Journal of the Americun Statistical Association, 71(33)):68 71, March 1976. ISSN 0162-1459. doi: 111110000112145.119977 194814477..
URL hhtpps:/www/
cttpe//ooninee.o/ooo/o/o/oooao/a 100.1111100000101111001111 1976 1048889117791111117788 10481478 10481478.
Publisher: ASA Website _eprin::
Gene Fliedner. Hierarchical forecasting: issues and use guidelines. Industrial Management & Data Systems, 101(1):5 12, February 2001. ISSN 0263-5577.
doi: 10.1105/0263557010333555 URL https://wwx
Rob J. Hyndman, Roman A. Ahmed, George Athanasopoulos, and Han Lin Shang. Optimal combination
forecasts for hierarchical time series. Compulational Statistics and Dala Analysis, 2011. ISSN 01679473.
ISBN: 0167-9473.
Jooyoung Jeon, Anastasios Panagiotelis, and Fotios Petropoulos. Probabilistic forecast reconciliation with applications to wind power and electric load. European Journal of Operational Research, 279 December 2019. ISSN 03772217. doi: 20.1016/j.ejor.2119.0.. cmt/rertreeree/11//8777711110000
http::://inkkiineien hwtp:/ghh.......eeeeeeeeeeeeeeee elsevier.
Aaron Lou, Isay Katsman, Qingxuan Jiang, Serge Belongie, Ser-Nam Lim, and Christopher De Sa. Differen- tiating through the Fréchet Mean. In Proceedings of the 37th International Conference on Machine Learn- ing, pp. 6393 6403. PMLR, November 2020. URL http:ttt//pcccceiiingggg//////et mlr wtwsss//119/ooo00...mm html ISSN: 2640-3498.
Spyros Makridakis, Evangelos Spiliotis, and Vassilios Assinakopoulos. M5 accuracy competition: Results, findings, and conclusions. International Journal of Forecnsting, 38(4):1346 1364, October 2022. ISSN 01692070. doi:
URL https://1inkinghu
pi1/80169207001001877
Jorge Nocedal and Stephen J. Wright. Numerical Oplimization. Springer Series in Operations Re- search and Financial Engineering. Springer New York, 2006. ISBN 978-0-387-3000--1 doi: 10.1007/ 978-0-387-40065-5. URL http::/link... wwwpndksspiinggee.oo wttp://ccoc/m.11000701111//01 11111100070778------000-55
Guy H. Orcutt, Harold W. Watts, and John B. Edwards. Data Aggregation and Information Loss. The Amer- ican Economic Review, 58(4):773 787, 1968. ISSN 0002-8282. URL https://www.jstor
1815532 Publisher: American Economic Association.
J.M. Ortega and W.C. Rheinboldt. Iterative Solution of Nonlinear Equations in Several Variables / SSAA Publications Labrary. Classics Applied Mathematics. STAM, 1970. URL htttps://epubr siam. org/
doi/book/10.
Anastasios Panagiotelis, George Athanasopoulos, Puwasala Gamakumara, and Rob J. Hyndman. Forecast reconelliation: A geometric view with new insights on bias correction. International Journal of Forecasting. 37(1):343 359, January 2021. ISSN 0070- doi: 1011066/.jforecsst2000..0
URL htlll:ttps////
22

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Anastasios Panagiotelis, Puwasala Gamakumara, George Athanasopoulos, and Rob J. Hyndman. Proba- bilistic forecast reconciliation: Properties, evaluation and score optimisation. European Journal of Op- erational Research, 306(2):693 706, April 2023. ISSN 0377-2217. doi: 10. 10.1016/jeejor:2222.ff
URL
2070. doi: 10. htt.://ccccccctcooooocccccccoaaa
Jeroen Rombouts, Marie Ternes, and Ines Wilms. Cross-temporal forecast reconciliation digital platforms with machine learning. International Journal of Forecosting, 41( 344, Jamuary 2025. ISSN 0169- 202.0.00....2.2..0000. URL
P11//00169200024000777
Evangelos Spiliotis, Mahdi Abolghasemi, Rob J. Hyndman, Fotios Pctropoulos, and Vassilios Assimakopou- los. Hierarchical forecast reconciliation with machine learning. Applied Soft Compuling, 112:107756,
November 2021. ISSN 1568-4946. doi: 2.0.00//...00.2222110170.11
URL bttps://ww...cieeoodiic..
Souhaib Ben Taieb, James W. Taylor, and Rob J. Hyndman. Hierarchical Probabilistic Forecasting of Eloctricity Demand With Smart Meter Data. Journal of the American Statistical Association, 116(533):
27 27-43, January 2021. ISSN 0162-1459, 1537-274X. doi: 10. 108000162111110011001111001000 400.4000..73600000 URL https:
wwww.tandfonline wwwplliiee.cmm.meeme http://....1111/.100////12222222 1080/006200000...11
1736081
J. A. Thorpe. Convex Surfaces. Tn A. Thorpe (ed.), Elementary Topics in Differential Geometry, pp- 95 107. Springer, New York, NY, 1979. ISBN 978-1-4612-6153-7. doi: 10.1007 URL http:////001.oog//0000000//11111
13.
Shanika L. Wickramasuriya, George Athanasopoulos, and Rob J. Hyndman. Optinal Forecast Reconcil- iation for Hierarchical and Grouped Time Series Through Trace Minimization. Journal of the Ameri- can Statistical Association, 114(526):804-819, April 2019. ISSN 0162-1459. doi: 10. 10100000122155..200 20111112111199111111111111111110 2018. 1448825
1448825. URL httggg/0.....0000000155511200110 wttpi/oorrrr111101..... https://doi.org/
Publisher: ASA Website eprint:
Daxhi Yang, Hao Quan, Vahid R. Disfani, and Carlos D. Rodriguez-Gallegoos Reconciling solar forecasts:
Temporal hierarchy. Solar Energy, 158:332-34 346, December 2017. ISSN 0038-092X, doi: 10.1016/j.solener 2017.09.055. URL https://w..sseenceiiiie.c
Lorenzo Zambon, Dario Azzimonti, and Giorgio Corani. Efficient probabilistic reconciliation of forecasts for real-valued and count time series. Statistics and Computing, 34(1):21, November 2023. ISSN 1573-1375. doi: (..107/s11122-0053
URL httpp:///do 00tt://0..000/111120000------000
A Appendix
A.1 Equivalence of problem equation 1 and FASE method
The FASE method (Brown Do Coutto Filho & De Souza 2009 formulates the state estimation procedure using a discrete time-variant state-space representation:
Tk+1 FE + 9k + Wk
(32)
yk h4 (xx) + Uk
(33)
Csually, in FASE, the state the magnitudes and angles of the voltages, while y represents the active and reactive powers. Equation 33 represents an a-priori forecast of the state x at time t+ 1 given the state estimation at time t. Nonlinear forecasting models could also be used, leading to variations of the method, but linear models have been found to be difficult to beat.
The objective of the FASE method is formulated as:
min || hx)) + || 1112
(34)
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where R and M are positive definite weighting matrices for the forecast errors in y and z, respectively.
To turn the nonlinear projection problem of ! into the FASE problem of 34 we define = 10", M = (f(=)) = 0} = ty = h(r)). Then 1 becomes:
min
|| Ell
(35)
s.t.
y h(x)
with weight matrix
M
W
R
The constraint h(z) implies that any feasible point must satisfy
h(x))
We can therefore rewrite the projection problem as an unconstrained optimization over æ:
f
min
[((x))
Expanding the norm with Block-diagone weights:
[ a 1 = |2 --1 + [1((2) -0l ((x)) 0 wlw
This is exactly the objective function of the FASE formulation. This shows the two objectives are equivalent when considering graph y h(x) for the definition of the manifold and when the weight matrix W is block-diagonal, partitioned according the dimensionality of and y.
A.2 Second fundamental form
Given f: R" - (z: [()= R" is a level set of f and a regular hypersurface f if ff(x) # on M. H =2 the Hessian of the function 2 Eg [=*((u-1) an orthonormal basis of the tangent space M,M Then the second fundamental form Ip IR at point M is symmetric bilinear form on the tangent space T,M, defined by:
T, w) = =D,u, 2, E T,M
(36)
v(2) Vf(22) [V](()]
(37)
where u(z) is the unit normal vector field, measures the rate of change of the normal vector the direction 2, Le., the normal curvature, In coordinates, this reduces to:
2 HT(pp ZET, [V
LD(2,2) 2)
(38)
which describes the change of the normal acceleration of the surface in the direction of x belonging to the langent space. A sub-level set is locally convex iff the second fundamental form is positive semidefinite (theorem and Thorpe 1979)):
(H(p) 2 for all T.M
(39)
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We can test this condition by inspecting if the Hessian projected on the tangent space is positive definite, equivalently, by checking if the smallest eigenvalue of the restricted Hessian Haan (p) is greater than 0:
Htan (p) = E, HD)
(40)
Amin (Httn(()) > 0 + (p)n(p)
(41)
where Ep E [ax((=1) an orthonormal basis of the tangent space TpM at p. This shows that Amin (Hean (p)) is a natural choice for inspocting the curvature of the levelset at p. Figure [1] shows different level sets for
(z) + 3 + z (left axis) and specific level set for f(z) z zf + z + z 221 (right axis), both colored by Amia tan) The second fundamental form be used write the following second-order approximation for M:
= (1) to (11113)
(42)
where i E T&M is in the tangent direction, while the correction introduced by the second fundamental form lies in the direction the unit normal v()), Im(Vf) C R".
A.2.1 Vector valued functions
In the case of manifold with codimension higher than 1, that is defined by a vector valued constraint f:R" + -R" M = {a R" f(z) = 0}. m < n, the normal space Im (Jy(z)) spans the Jacobian rows. this case (t, E Th is vector and the analogous expression equation 42 for second order approximation can be written as:
2 2 (E) (t((,) + (1141))
(43)
For the vector-valued case, orthonormal basis E for the tangent space T.M- ker J(z) be found by QR decomposition of the transposed Jacobian.
Level sets of f(z) x 3x7 + x3
#2)= x2 +x3
2x1x3=005
2.0
3.0
2.0
1.5
0.5
1.0
-1.0
0.5
0.5
0.0
0.5
0.0
1.0
-15
2.0
-1.
1.0
0.5
-15
-1.5
1.0
-0.5
3.0
0.5
1.0
-1.0
-1.5
-1.0
0.5
1.0
Min tigenvalue THE
Min tigenvalue TTe
Figure 11: Two examples of hypersurfaces colored by Amin Amin(Hran). These examples illustrate how Xmin (Htn)) can be thought of as the degree with which the surface bends away or bends towards a point in the ambient space.
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A.3 Proof of equation 12
In the following, we prove equation 12 for hypersurface M = {= f(z) 03 C R" with f having convex sub or super level set. Consider two points a and 2, both belonging to M and minimally displaced = 2-22 We can write the sccond-order Taylor expansion at 2:
0= f(z) f(3 f()) V)()) 1 " (()) + (1|11²)
(44)
Since f(2) 0, and since becomes tangent to M in the limit, neglecting higher order terms, write:
= (()ss E
(45)
where T&M is the tangent space Equivalently, we can write:
VS(3)* -= 8 -
(46)
where Fg E R is orthonormal basis of the tangent space M at E. Then:
sign (V)(=)²)) sign EE Es) -sign (Amin (HainnHannn (tan((
(47)
where the last equality comes from the definitencss of the restricted Hessian for convex / concave curves equation equation 41 We proved equation 47 in the limit 0, but since M is the boundary of convex set, it has a supporting hyperplane in 2, which means that sign (VS(z)²e) is constant on M, and thus equation 47 holds also for =
A.4 On the vacuousness instantaneous bounds for غ Y.
We want to bound the error |eme,t - ē that is, the distance from the estimated probability of improving RMSE after projection and the true probability or RMSE reduction. This can be decomposed via triangle incquality to consider the error of the Monte Carlo sampling:
le, Emo,tl S les - ēe + lē, E
The Monte Carlo estimation error M can be bounded easily as shown in lemma
Since Vi is never observed, bounding is more challenging. We can try to make explicit the dependence of T on the only observed sample Z4 Vt drawn from the true distribution time 4, using the decomposition:
lec Eel S lee
Y
+ E
Y
(48)
E
where Y; = h((r(Z))) is an indicator random variable, the evaluation of h((r(-)) at the observed sample at time t. This decomposition has the following interpretation: A represents the alcatoric uncertainty due to the unobservable distribution Pt, while E represents the epistemic uncertainty, which can made smaller by increasing the accuracy of the predicted law D. In the following we consider instantaneous bounds on these two quantities.
A: Unless produces degenerate distribution under he())(2)), the best possible global, determin- istic instantaneous bound for A is A < 1. That is, unless the true distribution is totally contained in one of the two half-spaces defined by 6i, the best instantaneous bound on A is uninformative since conditional (on the value of ex) deterministic instantancous
e, is a probability, bounded in 0, bound max(1 e2, 2 1/2. Furthermore, this bound approaches 1 when e, close to or 1, making it partieularly uninformative.
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E: The best possible instantancous bound on E is also uninformative, since it is the one that can be obtained when ē et. that is, the bound under perfect knowledge of the true distribution it, which is again max(1 - ez, ez).
Summing the two components results in a totally uninformative bound max(1 - ec, ex) 2 for ler Ezl therefore this specific decomposition can't be used bound T non-vacuously.
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Algorithm 1: Augmented Lagrangian nonlinear reconciliation with I.- IF-BFGS line search Input: forecast 2 I SPD weight W > constraint f : Rm
Input: tolerances tolfeas, tolgrad, tolstep, penalty parameters Po, Pmults Tine
Input: max iterations max_ outer, max inner
Whitening step: compute Cholesky W L L and set L-1
Set 1 + LE. Initialize yd < 9, - p0 + Po-
outer for h 0, 2,1.2.... max maa__oueer-a - do
Inner problem (unconstrained))
min Lx(y) = -1 (2F) + 2 ld dv) y(. yER"
Initialize gh: + y and L-BFGS state. for 0,,,j,,,--. mar inner do
Compute value and gradient ohi Lu(y""), ghd + ())(
Use zoom line search (satisfying Wolfe conditions) to compute L BEGS step Ay and update g + yhd Ayki.
if |لg I tolgrad or |Ayhill2 S tobtep then L break
Set ght1 yhss (final inner ilerate).
Recover gk+1 +- y and constraint d+1 f(zktl).
if ||ch+1|12 S tolmas then
L break
// outer stopping: feasibility reached
Multiplier update: A 2 + p Penalty update:
g+l
Pmuutt
if [|cc+1112
1|2 > Tine tolgass
otherwise.
return = as the reconciled forecast.
A.5 ALM LBFGS solver
This pscudocode describes an L-BFGS method equipped with a zoom line search enforcing Wolfe conditions, starting from the previous iterate. We terminate the inner loop when the gradient norm and step size are below fixed tolerances, and then update the multipliers and penalty parameter in a standard Augmented Lagrangian Method (ALM) fashion.
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A.6 Plots of tested manifolds
Paraboloid
MixedOuadratic
Exponeriial
208
109
A75
13
650
375
10 A
1 EM c2h 67%.05 130
LAS ant
Rastring
Rosen
160
ca 025 -033_0N
AbS
Himmeiblau
200
143
130
10
1.0
136
5.5
625
1 C58 .17 331
55
Figure 12: Plots of tested hypersurfaces.
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Quad-quad
quackog
22s2
10
130
C25
1.30
E30
4d-40
bowl-sin
Saddle-poly
335 OE
(25
rosenbrock
ring-trg
exp-cosh
0.5
60
130
-32
aT OCR JO33 2075
625
800
-314
Figure 13: Plots of tested manifolds with codimension 1. Since it's hard to visualize level sets of R4 R², plotted level sets of fi and fa separately.
30


Reviews:
[{"review_id": "89BercZvp5", "reviewer": "Reviewer_FCkM", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This research provides theoretical guarantees regarding whether nonlinear reconciliation, a method for transforming predictions to satisfy nonlinear constraints, actually reduces the Root Mean Squared Error (RMSE). Reconciliation is an important topic in practical situations where predictions are expected to satisfy appropriate constraints. However, there is not much research on the case with nonlinear constraints.\n\nThis study derives several theorems regarding the conditions under which reconciliation reduces the RMSE. First, Theorems 1 and 2 derive sufficient conditions for the RMSE to be improved by reconciliation for each given prediction point. However, subsequent experiments reveal that this condition is too optimistic. Therefore, Theorem 3 provides a method for numerically estimating the probability of improvement, rather than providing a binary indication of whether improvement occurs at each point. The usefulness and validity of the derived theorems are verified through detailed numerical experiments. This provides guidance on whether or not reconciliation should be performed.", "claims_and_evidence_yn": "Yes", "claims_explanation": "Basically, the paper is well written, and the claims are carefully examined by extensive numerical experiments. On the other hand, there are some concerns. I recommend making some improvements in this regard.", "audience_yn": "Yes", "audience_explanation": "Reconciliation is an important topic in machine learning and data analysis, and guaranteeing performance improvements under broader constraints is crucial. In this context, this work provides important insights related to reconciliation. Therefore, it is relevant to the readership of Transactions on Machine Learning Research.", "requested_changes": "# major points\n* My main concern is that without background knowledge on how reconciliation is used, the setup of Section 4. Numerical Tests will be hard to read. Theorems 1-3 abstractly deal with the effects of projecting onto a manifold, and can be understood independently. However, as soon as we get to Section 4.1, it becomes very specific, and readers may be confused by things like generating data using an AR model (Eq. 19) and performing training with LightGBM. Since TMLR readers cover a wide range of machine learning expertise, I think this is a major concern. At least, I was very confused when I first read it. Therefore, I recommend adding a paragraph or subsection in Section 4.1 that clearly explains what kind of data analysis problem is being addressed and how reconciliation is involved, in a way that is easy for those who have not had much experience with reconciliation to understand.\n\n# minor points \n* Regarding the notation in Theorem 3, it seems somewhat strange that $\\tilde{\\pi}_i$ remains in Eq. (15) after stating \"if $\\tilde{\\pi}_i = 1/k, \\forall i$\".\n* In Proof 3,  from (16) to (17), it approximates $\\nu(z)$ with $\\tilde{\\nu}(z)$, but is this generally valid? At least, I recommend adding some further discussion.\n* I interpreted that setting $\\tilde{\\pi}_i=1/k, \\forall i$ is possible when $\\tilde{z}_i$ are independent samples from $\\tilde{\\nu}$, but how easy is this generally? At least, I recommend adding some further discussion.\n\n## Very minor comments\n* On page 12, there is a statement that says \"a threshold of θ=5 or θ=6.\"  Isn't this a mistake, and should it be 0.5 or 0.6?", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "3cfRrmQ8nt", "reviewer": "Reviewer_YjXa", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper considered the problem of whether reconciliation could reduce the prediction error under nonlinear constraints. Two sufficient conditions are proposed for the single constraint case and for multiple constraints, respectively, under the assumption of simple geometry (the sub/suplevel set is convex/concave) of the constrained manifold. For general constrained manifold, the authors give an estimate of the probability that projecting the prediction onto the constrained manifold will reduce the error.", "claims_and_evidence_yn": "Yes", "claims_explanation": "The results in this paper are supported by proofs.", "audience_yn": "Yes", "audience_explanation": "People who are working on forecast reconciliation may be interesed in this work.", "requested_changes": "Overall, the paper generalize the existing works on reconciliation under linear constraints to nonlinear constraints. However, the writing of the paper needs to be significantly improved. In particular, many notations are used without a clear definition. The detailed comments are as follows. \n\n0. The author may further motivate the problem in the introduction. For example, why it is not a good choice to always do the projection since we already know the constraints? \n\n1. Many words are vague, e.g., \"nonlinear context\" in the abstract, \"linear case\" at the beginning of section 3. I suggest replace them by nonlinear constraints and linear constraints. \n\n2. The norm \\|  \\|_W should be defined. \n\n3. Just below Eq. (1). Is \"an (m-n)-dimensional differentiable manifold\" an assumption? \n\n4. The M defined for the linear reconciliation problem is not consistent in dimension. \n\n5. Eq. (4). Is the linear system uniquely solvable? \n\n6. Theorem 1. \\lambda_min is not defined. What is the meaning of \"restricted\"? It would be better to present Theorem 1 in a local manner, i.e., state the assumptions locally near the exact value and the prediction. \n\n7. Proo1. \\tilde \\delta is already defined and cannot be arbitrary. In addition, It is not clear what is the variable in \"\\nabla f(\\tilde z)^T \\tilde \\delta is a constant on M\". \n\n8. Section 3.2. Vector-valued functions might be replaced with multiple constraints for clarity. \n\n9. Section 3.3. What is the meaning of \"at multiple noise level\"? In addition, I think \\sigma_I should be somehow related to the magnitude of z_i. \n\n10. Example 1. M should be defined as M = {z = (x, y)^T \\in \\R^2: ...}. Is data D a subset of \\R^2? What are z_{1, i} and z_{2, i}? \n\n11. Theorem 3. k is not defined.", "broader_impact_concerns": "NA", "additional_comments": "NA", "extra_scores": {}, "extra_text": {}}, {"review_id": "yc0cdZ8Il7", "reviewer": "Reviewer_QAzm", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper studies nonlinear forecast reconciliation by projecting base forecasts onto a constraint manifold. It gives sufficient error-reduction conditions for (i) hypersurfaces with constant-sign curvature and (ii) higher-codimension manifolds, and proposes a Monte-Carlo “probabilistic test” driven by a reconciled predictive distribution. A Newton/KKT solver is outlined, and synthetic experiments illustrate when reconciliation helps.\n\nStrengths: \n- clear geometric setup\n- simple sufficient conditions\n- a practical Monte-Carlo test\n- simulations cover several toy manifolds.\n\nWeaknesses: \n- “probabilistic guarantees” amount to consistency of a plug-in estimator (no finite-sample/statistical guarantee under the true law)\n- the optimization block appears without convergence or existence/uniqueness conditions\n- empirical validation is synthetic only\n- some figures are hard to read.", "claims_and_evidence_yn": "No", "claims_explanation": "– The deterministic theorems are clean sufficient conditions, but only in the constant-curvature case, which is rather straightforward. The article fails to deliver meaningful results in the difficult cases.\n\n– The “probabilistic” part is a Monte-Carlo estimator under the modelled forecast law; the guarantee given is SLLN consistency, not a bound on expected error reduction under the true data-generating distribution.\n\n– Experiments are entirely synthetic with limited baselines; there is no real-data study to back practical relevance.", "audience_yn": "Yes", "audience_explanation": "Forecast reconciliation is widely used, and extending linear results to nonlinear constraints is of interest to forecasting and constrained learning communities. The geometric conditions and the simple decision rule are likely to attract readers, even if the empirical scope and guarantees need strengthening.", "requested_changes": "Critical :\n\n1) Calibrate the claims around “probabilistic guarantees.” Say clearly that the test estimates the probability of improvement under the forecast law; separate modelling assumptions from statistical ones.\n2) Add finite-sample bounds for the Monte-Carlo estimator (e.g., Hoeffding/Bernstein) and a simple sample-size guide for a target margin/CI.\n3) State well-posedness and solver guarantees: conditions for existence/uniqueness of the orthogonal projection and local/global convergence of the Newton/KKT updates.\n4) Clarify the role of W in Eq. (1). If results require W=I, say so; otherwise sketch the extension or move W≻0 to discussion.\n5) Fix structure: relocate §2.1 next to the probabilistic test or add forward references so the flow is coherent.\n\nNon-critical :\n\n6) Improve figures.\n7) Add at least one real-data benchmark.", "broader_impact_concerns": "NA", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)