Functional Equivalence and Path Connectivity of Reducible Hyperbolic Tangent Networks
Authors: Matthew Farrugia-Roberts
Venue: NeurIPS 2023 poster (2023)
Keywords: theory
neural network theory
structural redundancy
functional equivalence
functional equivalence class
partial identifiability
parameter canonicalisation
parameter space
piecewise-linear
connectivity
TL;DR: By accounting for various redundant arrangements of units, we characterise all different neural net parameters that can implement any given function, and we show piecewise-linear connectivity properties of these sets (architecture: shallow tanh nets)

Abstract: Understanding the learning process of artificial neural networks requires clarifying the structure of the parameter space within which learning takes place. A neural network parameter's functional equivalence class is the set of parameters implementing the same input--output function. For many architectures, almost all parameters have a simple and well-documented functional equivalence class. However, there is also a vanishing minority of reducible parameters, with richer functional equivalence classes caused by redundancies among the network's units.

In this paper, we give an algorithmic characterisation of unit redundancies and reducible functional equivalence classes for a single-hidden-layer hyperbolic tangent architecture. We show that such functional equivalence classes are piecewise-linear path-connected sets, and that for parameters with a majority of redundant units, the sets have a diameter of at most 7 linear segments.

functions. Thus, Functional Equivalence and Path Connectivity of Reducible Hyperbolic Tangent Networks
Matthew Farrugia-Roberts School of Computing and Information Systems The University of Melbourne matthew&far: net
Abstract
Understanding the learning process of artificial neural networks requires clarify- ing the structure the parameter space within which learning takes place. A neural network parameter's functional equivalence class is the set of parameters implementing the same input-output function. For many architectures, almost all parameters have simple and well-documented functional equivalence class. However, there is also a vanishing minority reducible parameters, with richer functional equivalence classes caused by redundancies among the network's 's units.
In this paper, we give an algorithmic characterisation of unit redundancies and reducible functional equivalence classes for a gent architecture. We show that such functional equivalence classes are piecewise- linear path pathecon sets, and that for parameters with majority of redundant units, the sets have diameter of at most linear segments.
hyperbolic tan-
Introduction
Deep learning algorithms construct neural networks through local search in a high-dimensional parameter space. This search is guided by the shape of some loss landscape, which is in turn de- termined by the link between neural network parameters and their understanding the link between parameters and functions is key to understanding deep learning.
It is well known that neural network parameters often fail to uniquely determine an input-output function. For example, exchanging weights between two adjacent hidden units generally preserves functional equivalence -Hecht-Nielsen, 1990). For many architectures, almost all parameters have simple class of functionally equivalent parameters. These classes have been characterised for multi- layer feed-forward architectures with various nonlincarities (e.g., Sussmann. 1992: Albertini et al., 1993; Kūrková and Kainen, 1994; Phuong and Lampert, 2020; Vlačíé and Böleskei, 2021).
However, all existing work on functional equivalence excludes from consideration certain measure sets of parameters, for which the functional equivalence classes may be richer. One such family of parameters is the so-called reducible parameters. These parameters display certain structural redundancies, such that the same function could be implemented with fewer hidden units (Sussmann, 1992; Vlačić and Bölcskei, 2021), leading to richer functional equivalence class.
Despite their atypicality, reducible parameters may play important role in deep learning. Learning exerts non-random selection pressure, so measure zero sets parameters may still arise in practice, and indeed reducible parameters 2023). These parameters are also a source of information singularities (cf. Fukumizu, 1996), relevant statistical theories of deep learning (Watanabe, 2009; Wei et al., 2022). Moreover, the structure of functional equivalence classes has implications for the local and global shape the loss landscape, in ways that may influence learning dynamics near reducible parameters.
appealing solutions due parsimony (cf. Farrugia- Farrugia-Roberts, Roberts,
37th Conference on Neural Information Processing Systems (NcurIPS 2023).

---

In this paper, we study functional equivalence classes for Singlehhideen-aa networks with the hy- perbolic tangent nonlinearity, building on the foundational work of Sussmann (1992) on reducibility in this setting. We offer the following theoretical contributions.
1. In Section 4, we give a formal algorithm producing a canonical representative parameter from any functional equivalence class, by systematically eliminating all sources of struc- tural redundancy. This extends prior algorithms that only handle irreducible parameters.
2. In Section 5, we invert this canonicalisation algorithm to characterise the functional equiv- alence class of any parameter as a union of simple parameter manifolds. This characterisa- tion extends the well-known result for irreducible parameters.
3. We show that in the reducible case, the functional equivalence class is piecewise- path-connected set-that is, any two functionally equivalent reducible parameters are con- nected by a piecewise linear path comprising only equivalent parameters (Theorem 6.
4. We show that if a parameter has a high degree of reducibility (in particular, if the same func- tion can implemented using half of the available hidden units), then the number of linear segments required to connect any two equivalent parameters is at most 7 (Theorem 6.3).
While the single-hidden-layer hyperbolic tangent architecture is not immediately relevant to modern deep learning. it enables the first comprehensive analysis of neural network structural redundancy. Moreover, feed-forward layers are a fundamental building block of many modern architectures, and so our analysis partially informative for such extensions. In Section 7 we discuss such extensions and others, as well as connections to other topics in deep learning including loss landscape analysis, model compression, and singular learning theory.
2 Related Work
Sussmann (1992) studied functional equivalence in single-hidden-layer hyperbolic tangent networks, showing that two irreducible parameters are functionally equivalent if and only if they are related by simple operations of exchanging and negating the weights of hidden units. This result was later extended to architectures with broader class of nonlincarities (Albertini et al., 1993; Kürková and Kainen, I 1994), to architectures with multiple hidden layers (Fefferman and Markel, 1993; Fefferman, 1994), and to certain recurrent architectures (Albertini and Sontag, 1992, 1993a,b.c). More recently, similar results have been found for ReLU networks (Phuong and Lampert, 2020; Bona-Pellissier et al., 2021; Stock and Gribonval, 2022), and Vlačić and Böleskei (2021, 2022) have generalised Sussmann's 's results to a very general class of architectures and nonlincarities. However, all of these results have come at the expense of excluding from consideration certain measure zero subsets of parameters with richer functional equivalence classes
A similar line of work has documented the global symmetries of the parameter space-bulk bulk transfor- mations of the entire parameter space that preserve all implemented functions. The search for such symmetries was launched by Hccht-Nielsen (1990). Chen et al. (1993, also Chen and Hecht-Nielsen, 1991) showed that in the case of multi-layer hyperbolic tangent networks, all analytic symmetries generated by unit exchanges and negations. Rüger and Ossen (1997) extended this result to addi- tional sigmoidal nonlinearities. The analyticity condition excludes discontinuous symmetries acting selectively on, say, reducible parameters with richer equivalence classes (Chen et al., 1993).
Rüger and Ossen (1997) provide a canonicalisation algorithm. Their algorithm negates each hidden unit's 's weights until the bias is positive, and then sorts each hidden layer's units into non-descnn order by bias weight. This algorithm is invariant precisely to the exchanges and negations mentioned above, but fails to properly canonicalise equivalent parameters that differ in more complex ways.
To our knowledge, there one line of work bearing directly on the topic of the functional equiv- alence classes of reducible parameters. Fukumizu and Amari (2000) and Fukumizu et al. (2019) have catalogued methods of adding a single hidden unit to a neural network while preserving the network's 's function, and Şimşek et al. (2021) have extended this work to consider the addition of multiple hidden units. Though derived under a distinct framing, it turns out that the subsets of pa- rameter space accessible by such unit additions correspond to functional equivalence classes, similar to those we study (though in a slightly different architecture). We note these similarities, especially regarding our contributions (2) and (3), in Remarks 5.4 and 5.5 and Remark 6.2.

---

3 Preliminaries
We consider family of fully-com cully-connected, weed-forward neural network architectures with a single input unit, single biased output unit, and a single hidden layer of h E N biased hidden units with the hyperbolic tangent nonlinearity tanh(2) (2) (e* e //(e + e=*). Such an architecture has parameter space W₁ RM++1 Our results generalise directly networks with multi-dimensional inputs and outputs, as detailed in Appendix A.
The weights and biases of the network's units encoded in the parameter vector in the format (01, b1, Ci..... ah bn, ch. d) E Wn where for each hidden unit i = 1,.... h there is an outgoing weight a; E I. an incoming weight b, E I,, and bias C2 E IR, and d E IR is an output unit bias. Thus each parameter w (a1, b1, C1y..., fw IR
.......-0h Oh bn, Ch d) E W, indexes a mathematical function
R defined as follows:
(=( = d+ Σ tanh(bar + ci).
Two parameters w E Wh w E Wh are functionalll equivalent if and only if fw fwr as functions on R (that is, Vx IR, (z) fx)()))) Functional equivalence is of course equivalence relation on Wn, Given a parameter w E W the functional equivalence class of w, denoted & [w), is the set of all parameters in W₁ that functionally equivalent to w:
8 [w] = { w E Wn Ja = Sw'
For this family of architectures, the functional equivalence class of almost all parameters is a discrete set fully characterised by simple unit negation and exchange transformations d;, Ti, Wn, + Wn. for i 1,.... h. where
(an, bi, ... Oh, bn, Ch, d) = (ai, bi, Cl, Gi bi Ci ah bns Ch, d) Ti. (a1, bi, C1, ................................ an, bh, Ch. d) (a1, b,, C.........--............ C1-1, C1, Dj a...jbbbb. bj, Cj. 0i+1,
Cj-1, 0g bi, c, aj+1. an, bu, Ch. d)-
More formally, these transformations generate the full functional equivalence class for all so-called irreducible parameters (Sussmann, 1992). A parameter w = (a), b1, Cr,..., ah, bn, Ch. d) E Wn, is reducible if and only if it satisfies any of the following conditions (otherwise, w is irreducible):
(i) a; = 0 for some i, or
(ii) b. = for some i, or
(iii) (bs, c.) = (b,, cj) for some + j.
(iv) (b,,c;) = (-b -c,) for some + j.
Sussmann (1992) also showed that in this family of architectures, reducibility corresponds to non- minimality: parameter
W,, is reducible if and only if w is functionally equivalent to some w' E W2 with fewer hidden units h' < h. We define the rank of w, denoted rank(u) w), as the minimum number of hidden units required implement fw:
rank(w) = min { h E N Eu E Wwi fa = fw
Finally, we make use of the following notions of connectivity for set of parameters. Given a set W C Wn, define a piecewise linear path in W as continuous function p [0, 1] > comprising a finite number of linear segments. Two parameters w, w' E W, are piecewise- -ineer path-connected in W, denoted w *** w' (with implicit), if there exists piecewise linear path in W such that p(0) w and = w'. w'. Note that ewe is equivalence relation on W. A set W C W, is itself piecewise-lined path- math-conmected if and only if ewo is the full relation, that is, all pairs of parameters in W are piecewise linear path-connected in W.
The length of a piecewise linear path the number of maximal linear segments comprising the path. The distance between two piecewise linear path-connected parameters is the length of the shortest path connecting them. The diameter of piecewise linear path-connected set is the largest distance between any two parameters in the set.
3

---

Parameter Canonicalisation
parameter canonicalisation algorithm maps each parameter functional equivalence class to a consistent representative parameter within that class. A canonicalisation algorithm serves as theoretical test of functional equivalence. In Section we invert our canonicalisation algorithm to characterise the functional equivalence class. More practically, canonicalising parameters before measuring the distance between them yields metric that independent of functionally irrelevant details such as unit permutations.
Prior work has described canonicalisation algorithms for certain irreducible parameters (Rüger and Ossen, 1997); but when applied to functionally equivalent reducible parameters, such algorithms may fail to produce the same output. We introduce canonicalisation algorithm that properly canon- icalises both reducible and irreducible parameters, based on similar negation and sorting stages, combined with novel reduction stage. This stage effectively removes 'zeroes out' redundant units through various operations that exploit the reducibility conditions. This process isolates functionally equivalent but irreducible subparameter.
Algorithm 4.1 (Parameter canonicalisation). Given a parameter space WR proceed:
procedure CANONICALISE(w (01, b1, CI..... an bb, Cha d) W;)
2: Stage 1: Reduce the parameter, zeroing out redundant hidden units
3: Z x (}
keep track of 'zeroed' units
4: while any of the following four conditions hold do if for some hidden unit & Z, a; = then
5:
b reducibility condition (i)
6:
bg.c < 0
7:
+ Z (+}
8:
else if for some hidden unit & Z, b 0 then
(ii)
9:
4 d a, Lanh(c)
10:
dg + 7 U {+}
+ 0
11:
12:
else if for some hidden units i,j $ Z, ",i (b., G. (b), then
(iil)
13:
aj - 4
14:
a, b,,
15:
Z + Z {c} else if for some hidden units i,j $ ZZi+ (b,, = (-bj,- then
16:
(iv)
17:
4j < aj
18:
0i bi, G
19:
2 + Z U {i}
end
21:
end while
Stage Negate the nonzero units have positive incoming weights
23:
for each hidden unit & ZZ do
a; b,. + sign(b;) (a,, b;, ci) 25: end for
Stage 3: Sort the units by their incoming weights and biases
27: + permutation sorting i .. by decreasing b; breaking ties with decreasing Ca w + (dm(1), (x))); Cm(1): ..., (x))); bx( (a()): d)
29:
Now, w. has been mulated into the canonical equivalent parameter
30:
return w
31:
end procedure
The following theorem establishes the correctness of Algorithm 4.1.
Theorem 4.2. Let w, w' E W₁, Let CANONICALISE(wu) and v' CANONICALISE (w"). Then (i) u is functionally equivalent to w; and (ii) if w and w' are functionally equivalent, then v = v'.
Proof. For (i). observe that fw maintained by each iteration of the loops Stages 1 and and by the permutation in Stage 3. For (ii). observe that Stage I isolates functionally equivalent and irreducible subparameters u E W, and u' E W,, of the input parameters w and w' (excluding the zeroed units). We have fu = f. Sar fur, so by the results of Sussmann (1992), rank(w), and and w' are related by unit negation and exchange transformations. This remains true in the presence of the zero units. Stages 2 and 3 are invariant precisely such transformations. D

---

Full Functional Equivalence Class
Algorithm 4.1 produces consistent output for all parameters within a given functional equivalence class. serves as the basis for the following charactcrisation of the full functional equivalence class.
The idea behind the characterisation is to enumerate the various ways for parameter's 's units to be re- duced, negated, and sorted throughout Algorithm 4. Each such canonicalisation trace corresponds to simple set of parameters that takes exactly this path through the algorithm, as follows.
Definition 5.1 (Canonicalisation trace). Let h E N, S h. A canonicalisation trace of order on h units is a tuple (a, T), where a E {-1, +1 h is a sign vector (interpreted as tracking unit negation throughout the algorithm); and - .1.....h) + {0, .......,, h is a function with range including {I..... r (interpreted as tracking unit reduction and permutation throughout the algorithm).
Theorem 5.2. Let w E W and v = (a1, 31, ",...... ,r,, Br.hh. 5) = CANONICALISE((U)) Let rank(w) w). Then the functional equivalence class &
Th,
g(w] W,, is union of subsets
(x²² [1o n m y
h. O Nn AZ.
8(w] =
U
(1)
(a.T)E((b.))
==1
1=++
where [h, r) denotes the set of all canonicalisation traces of order r on h units and )
E 1,b, = and
x²-
(a1, b1, ar: bn, Ch. d) Wh
d + Σier as tanh(c)) - f (B,7)
Vi I, (bi,
(a1, (1,b1,,,,,.. an bn ch. d) E Wh
and Eier 0;0; = f and
Vi.j E 1,0, (b,, ci) = og (b), cj)
Z (a1, b1.
C1, (1,,bi,1,....
an. bn, Ch. d) E Wn.
and Σiel O2a,- = 0
Proof. Suppose w' = (d1, b/1 c...., oh S/ che d') E W₁ is in the union in (1), and therefore in the intersection for some canonicalisation trace (a, r(h, Then fwr fo swo as follows:
(x) d' + Σ of tanh(b62x + c) + ΣΣ df tanb(bke + +c) + Σ Σ at tan + c)
iEr [0]
1=1 i
= + aj tanh(3,a; 73) since w' x
ya o, j=1
j=++1
Now, suppose w' E 3 [w]] Construct canonicalisation trace (o, E I(hh r) following the execu- tion of Algorithm 4.1 on w'. Set 0, = -1 where sign(b)) = =1, otherwise -1. Construct 7 from identity follows. In each Stage 1 iteration, if the second branch is chosen, remap r(i) to O. If the third or fourth branch is chosen, for & E F --) (including itself), remap r(k) to j. Finally, incorporate the Stage permutation क: simultaneously for k & T T0) remap T Tlk) to x (T())))
Note CANONICALISE (w') = v by Theorem 4.2. Then w' E X +-1 because (0) contains exactly those units incorporated into 8. Moreover, for = 1...., w/ E y because
T [-] contains exactly those units incorporated into unit j v, and a their relative signs (3, > 0).
Likewise, for j E 1...., h, w' E Za.r-10 (which is vacuous if 7- [5] is empty).
Remark 5.3. If w E W, irreducible, then rank(w) (w) h. For (a, r) E [(h, 7 is permutation
(since the range must include (1....,h} The set of traces therefore corresponds to the set of transformations generated by unit negations and exchanges, as in Sussmann (1992).
Remark 5.4. When rank(w) (w) = h 1, there are, modulo sign vectors and permutations, essentially three canonicalisation traces, corresponding to the three ways of adding an additional unit (h - 1) -unit network discussed by Fukumizu and Amari (2000) and Fukumizu et al. (2019): to introduce new constant unit one with zero output, or split existing unit in two.
Remark 5.5. Similarly, in Şimşek et al. (2021, Definitions 3.2 and 3.3), an (r + j)- coupled with permutation plays the role of in characterising the expansion manifold, akin to the functional equivalence class but from the dual perspective of adding units to an irreducible parameter. Şimşek et al. (2021) study setting without unit negation symmetry, there is need for a sign vector.
5

---

Path Connectivity
In this section, show that the reducible functional equivalence class is piecewise linear path- connected (Theorem 6.1), and, for parameters with rank at most half of the available number of hidden units, has diameter at most 7 linear segments (Theorem 6.3).
Theorem 6.1. Ler w E Wn If w is reducible, then [(w] piecewise linear path-connected.
Proof. It suffices to show that each reducible parameter E Wp is piecewise linear path-connected in B[w] to its canonical representative CANONICALISE((W) The path construction proceeds by tracing the parameter's mutations in the course of execution of Algorithm 4.1. For each iteration of the loops in Stages and 2. and for each transposition in the permutation in Stage 3, we construct a multi-segment sub-path. To describe these sub-paths, each sub-path as w = (a1, bj CI...., ag. bn, Ch2 d), noting that this parameter is mutated throughout the algorithm, but is functionally equivalent to the original at all of these intermediate points.
denote the parameter at the beginning of
1. In each iteration of the Stage loop, the construction depends on the chosen branch, as follows. Some examples illustrated in Figure 1.
(i A direct path interpolating b; and o; to zero.
(ii) A two-segment path, interpolating to zero and d to d + tanh(c:), then e; to zero.
(iii A two-segment path, interpolating to zero and a, to aj + 04, then b, and to zero.
(iv) A two-segment path, interpolating to zero and aj to aj ag then by and e to zero.
(ii)
(iii)
Tbi,0
(iv)
bi.
8 [p]
by:cy
(ur
b
8 [u]]
w
a
04
m
Figure I:
Example paths constructed for each of the Stage I branches. Other dimensions held fixed.
Since (the original) w is reducible, (the current) w must have gone through at least one iteration in Stage 1, and must have at least one blank unit l with aks by, Ck O. From any such parameter w, there is a three-segment path in 8 [(wo] that implements a blank-exchange manoeuvre transferring the unit k, and leaving 04, by, G. 0: first interpolate by to b, and G₁ to e; then interpolate ag to a; and to zero; then interpolate 6; and e; to zero. Likewise, there is sereessegment path that implements a negative blank-esd cannk-exchange manoeuvre, negating the weights as they are interpolated into the blank unit. With these manoeuvres noted, proceed:
weights of another unit
2. In each iteration of the Stage 2 loop for which sign(b,) = -1, let k be a blank unit, and construct a six-segment path. First, elank-exchange unit into unit k. Then, negative blank-exchange unit & into unit i. The net effect is to negate unit i.
3. In Stage 3, construct a path for each segment in a decomposition of the permutation as product of transpositions. Consider the transposition (i, j). If or j is blank, simply blank- exchange them. If neither is blank, let & be a blank unit. Construct nine-segment path, using three blank-exchange manoeuvres, using as 'temporary storage' to implement the transposition: first blank-exchange units and k then blank-exchange units (now blank) and j. then blank-cxchange units j (now blank) and & (containing i's original weights).
The resulting parameter is the canonical representative and it can be verified that each segment in each sub-path remains in $ [w] as required.
Remark 6.2. Şimşek et al. (2021, Theorem B.4) construct similar paths to show the connectivity of their expansion manifold (cf. Remark 5.5). They first connect reduced-form parameters using blank- exchange manoeuvres and then show inductively that each unit addition preserves connectivity.

---

Theorem 6.3. Let w E y. rrank((w) - 2' then [(u] has diameter at most 7.
Proof. Let W₁ with rank(w) r 2. Let w/ E 8(w]. We construct piecewise linear path from w to w' with 7 segments. By Theorem 6.1, path exists via the canonical representative parameter & = CANONICALISE((e)) However, this path has excessive length. We compress the length to 7 by exploiting the following opportunities to parallelise segments and 'cut corners'. These optimisation steps are illustrated in Figure
(a) Let the Stage result from Algorithm 4. for w be denoted u. Let the Stage I result for w' be denoted u'. Instead of following the unit negation and exchange transformations from to v, and then back to n', we transform u into u' directly, not (necessarily) via v.
(b) We connect w to u using two segments, implementing all iterations of Stage in parallel. The first segment shifts the outgoing weights from the blank units to the non-blank units and the output unit bias. The second segment interpolates the blank units' incoming weights and biases zero. We apply the same optimisation to connect w' and u'.
(c)
We connect u and u' using two blank-exchange manoeuvres (6 segments), exploiting the majority of blank units as 'temporary storage'. First, we blank-exchange the non-blank units of into blank units of u', resulting parameter ū' sharing non-blank units with u'. Then, we (negative) blank-exchange those weights into the appropriate non-blank units of u', implementing the unit negation and exchange transformations relating u, i' and u'.
(d) The manoeuvres in (b) and (c) begin and/or end by interpolating incoming weights and biases of blank units from and/or to zero, while the outgoing weights are zero. We combine adjacent beginning/end segments together, interpolating directly from the start to the end, without (necessarily) passing through zero. This results in the required seven-segment path, tracing the sequence of parameters w, w w² ..... w w/ E Wn.
w
w
(b) & (c)
w
u
V..
AZzz
4
Zio 25 www.rronn
(d)
w
w
4
ur3
w
path before optimisation
1/
path after optimisation
path unchanged by optimisation
w2
w.
Figure 2: A conceptual illustration of the four path optimisations, producing a seven-segment piece- wise linear path of equivalent parameters in high-dimensional parameter space. (a) Follow unit negation and exchange transformations directly between reduced parameters, not via the canonical parameter. (b) & (c) Parallelise the reduction steps, and use the majority of blank units to parallelise the transformations. (d) Combine first/last segments of reduction and blank-exchange manoeuvres.
To describe the constructed path in detail, we introduce the following notation for the components of the key parameters w, w', U. u', wt, w² s..., w6 E W
(a"", by: of" ...., an by Ch. d w' = (at' w/
(a) by ct ..... ans Dh.
Ch. d')) Dy u de
b w' On by w _
n' = (af D o
& = (a U4, ....., ag U/, di, d²))
(k = 1.....6) 6).
Of the units u, exactly r are blank those in the set from CANONICALLSSE(w). Denote the complement set of non-blank units U = (1,...,A (....,)) .,) } \ Z. Likewise, define Z' and U/' from v'.
7

---

With notation clarified, we can now describe the key points w .o......." w in detail, while showing that the entire path is contained within the functional equivalence class [w].
1. The first segment interpolates each outgoing weight from a to a, and interpolates the output bias from d' to d² That is, w = (at: , :...., :....
dg b% C d"))
To see that this segment is within 8 fww], observe that since the incoming weights and biases are unchanged between the two parameters, +t=*+(110)w222
(L (1) + (1 fm (x) for x E RR and t E [0. 1]. To show that fu = fm' we construct a function T {1..... h} 7 {0, .,..,...hh) from identity following each iteration of Stage of CANONICALISE( w): when the second branch is chosen, remap r() to O; and when the third or fourth branch is chosen, for E -i] (including i itself), remap T rlk) to j. Moreover, we define sign
vector E {1, where o; = -1 if sign(bş) = -1, otherwise o; = +1. Then:
(w) = d + +E - Eier M of tanh(b," ci"))
= =d + Σ 1o at tanh (c,") + Ei= (E (EEr-iy tanh x + cg))
= d E dz tanh(0 + cgg fw)
2. The second segment completes the reduction and begins the first blank-exchang ma- noeuvre to store the nonzero units in Zz. For U n U', pick distinct 'storage' units enough,
r- 12-Inuu nuI S. (h - r) IZnu" by assumption thus U n U" U - -Z n n - n = I' n Z|. Interpolate unit j's incoming weight from bj" to b." and interpolate its bias from oj to ci" Meanwhile, for all other j E Z, interpolate the incoming weight and bias to zero. This segment is within [[w] as for j Z, a aj by definition of Z.
j E Z n Z'. There
3. The third segment shifts the outgoing weights from the units in U'U' to the units in Znz' prepared in step (2). For E U U', pick the same storage unit as in step (2). Interpolate unit j's outgoing weight from a = to a; and interpolate unit i's outgoing weight from at to zero. This segment is within 3 [w] as b2 = 03 and o = of by step (2).
4. The fourth segment completes the first blank-exchange manoeuvre and begins the second, to form the units of u'. For E U", interpolate unit i's incoming weight from by to by' and interpolate its bias from of to c²' This segment is within 8(w) because for i E U i Z.
ai a. = 0 by definition of Z, and for i U' n U, at = 0 by step (3).
5. The fifth segment shifts the outgoing weights from the selected units in Z' the units in U prepared in step (4). We simply interpolate each unit i's outgoing weight to a.'
To see that the segment is within 0 [w]. note that u and w' are related by some unit negation and exchange transformations. Therefore, there is correspondence between their sets of nonzero units, such that corresponding units have the same (or negated) incoming weights and biases. Due steps (2)-(4) there are 'storage' units in w with the weights of the units of u. and the correspondence extends to these storage units. Since the storage units are disjoint with U', this fifth segment has the effect of interpolating the outgoing weight of each of the storage units j E Z' in w" from a. to zero (where is as in step (3)), while interpolating the outgoing weight of its corresponding unit k U' from zero to I oj (where the sign depends on the unit negation transformations relating u and u').
6. The sixth segment completes the second blank-exchange manoeuvre and begins to reverse the reduction. For E Z', interpolate unit i's incoming weight from &e to bt" and interpo- late its bias from cf to ct This segment is within T [w] as for i E Z', aa =a by definition of Z'.
7. The seventh segment, of course, interpolates from 6e to w'. To see that this segment is within G [[w]. note that by steps (5) and (6), w = (af' by' .er' ...., a% by ooy d'') (noting dd = d' since the output unit's bias preserved by unit transformations). So the situation is the reverse of step (I), and similar proof applies.
8

---

7 Discussion
In this paper, we have investigated functional equivalence classes for reducible single-hidden-layer hyperbolic tangent network parameters, and their connectivity properties. Recall that irreducible parameters have discrete functiinal equivalence classes described by simple unit negation and ex- change transformations. In contrast, reducible functional equivalence classes form a complex union of manifolds, displaying the following rich qualitative structure:
There is a central discrete constellation of reduced-form parameters, each with maximally many blank units alongside an irreducible subparameter. These reduced-form parameters are related by unit negation and exchange transformations, like for irreducible parameters.
Unlike in the irreducible case, these reduced-form parameters are connected by network of piecewise linear paths. Namely, these are (negative) blank-exchange manoeuvres, and, when there are multiple blank units, simultaneous parallel blank-exchange manoeuvres.
Various manifolds branch away from this central network, tracing in reverse the various reduction operations (optionally parallel). Dually, these manifolds trace methods for adding units (cf., Fukumizu and Amari, 2000; Fukumizu et al., 2019; Şimşek et al., 2021),
Theorem 6.3 establishes that when there is majority of blank units, the diameter of the entire union of manifolds becomes a small constant number of linear segments. With fewer blank units it will sometimes require more blank-exchange manoeuvres to traverse the central network of reduced- form parameters. Future work could investigate efficient implementation of various permutations through parallel blank-exchange manoeuvres with varying numbers of blank units available.
Towards modern architectures. Mingle-hidden-layee hyperbolic tangent networks appear far from relevant to modern deep learning architectures. However, we expect our canonicalisation algo- rithm to partially generalise and some aspects of our connectivity findings to generalise as follows.
Structural redundancy. Observe that reducibility conditions (ii- apply to structural redundancies that are generic to every feed-forward layer within any architecture (units with zero, constant, or proportional output). Unit negation symmetries are characteristic of odd nonlinearities only, but other nonlinearities will exhibit their own affine symmetries that would play a similar role. more sophisticated architectures, these basic sources of structural redundancy will sit alongside other sources such as interactions between layers, attention blocks, layer normalisation, and so on.
Canonicalisation. It follows that Algorithm 4.1 serves partial canonicalisation for more sophisti- cated architectures (with small modifications for alternative nonlinearities). Future works need only find and remove the additional kinds of structural redundancy.
Connectivity. Similarly, additional sources of redundancy expand the functional equivalence class. While global connectivity properties may be lost in this process, individual paths will not be dis- turbed. We expect that our high-level qualitative finding will hold: that the more reducible param- eter is, the more intricately connected it is to functionally equivalent parameters.
Towards approximate structural redundancy. Our framework is built around the definition of functional equivalence, which requires exact equality of functions for all inputs. A more pragmatic definition would concern approximate equality of functions for relevant inputs. One step in this direction is studying proximity in parameter space to low-rank parameters, though detecting such proximity precisely formally intractable Roberts,-Roberts. 2023).
Functional equivalence and the loss landscape. Functionally equivalent parameters have equal loss. Continuous directions and piecewise linear paths within reducible functional equivalence classes therefore imply flat directions and equal-loss paths in the loss landscape. More broadly, the set of low- zero-loss parameters is union of functional equivalence classes. If some (very) reducible parameters obtain low loss (such as in the overparameterised setting) then the set of low- loss parameters contains rich functional equivalence classes subsets.
Of course, having the same loss does not require functional equivalence. Indeed, Garipov et al. (2018) observe functional non-equivalence in low-loss paths. The exact relevance of reducible pa- rameters to these topics remains to be clarified. Of special interest the connection to theoretical work involving unit pruning (Kuditipudi al., 2019) and permutation symmetries (Brea et al., 2019).

---

Canonicalisation and model compression. Our canonicalisation algorithm transforms neural network parameter into another parameter that has the same ontut behaviour but effectively uses fewer units. This size reduction is not fundamental to the goals of canonicalisation (we could still achieve canonicalisation by producing a standard, maximally dense representation of each equivalent parameter). Nevertheless, Algorithm 4.1 performs (lossless) model compression as side-effect (ef. Farrugia-Roberte 2023).
In particular, our canonicalisation algorithm is reminiscent of a unit-based pruning technique (cf., e.g,, Hoefler et al., 2021). However, there are few salient differences. First, pruning algorithms are usually not designed to preserve exact functional equivalence on all inputs, but rather to accept small changes in outputs (on specific inputs). Second, unit-based pruning techniques usually select individual units for removal, and then continue training the network. Our canonicalisation algorithm instead considers operations that simultaneously remove a unit and perform systematic adjustment to the remaining weights and biases to maintain the network's 's behaviour without any training.
Of interest, Casper al. (2021) empirically studied a network pruning that finds units with weak outputs or pairs of units with correlated outputs, and then eliminates or merges these units and makes appropriate adjustments approximately maintain performance. The elimination and merging op- erations bear striking resemblance to the operations in Algorithm 4.1.
Reducible parameters and singular learning theory. When a reducible parameter is a critical point of the loss landscape, it is necessarily degenerate critical point (due to the continuum of equal- loss equivalent parameters nearby; or cf. Fukumizu, 1996). This places situations where learning encounters (or even approaches) reducible parameters within the domain of singular learning theory (cf. Watanabe, 2009; Wei et al., 2022).
The nature of the degeneracy depends on the exact ways in which it is reducible. For example, is the parameter at the intersection of multiple of the manifolds comprising the functional equivalence class (cf. Theorem 5.2)? The order of variation in directions away from the functional equivalence class also plays a role (cf. Lau et al., 2023). Future work could analyse the structural redundancy to find principled bounds effective dimensionality.
Conclusion
Reducible parameters exhibit structural redundancy, in that the same onput-output function could be implemented with smaller network. While reducible parameters comprise measure zero subset of the parameter space, their functional equivalence classes are much richer than those of irreducible parameters. Understanding these rich functional equivalence classes is important to understanding the nature the loss landscape within which deep learning takes place.
We have taken the first step towards understanding functional equivalence beyond irreducible param- eters by accounting for various kinds of structural redundancy in the setting of hinllehhidden-lyerr hyperbolic tangent networks. We offer an algorithmic characterisation of reducible functional equiv- alence classes and an investigation of their piecewise linear connectivity properties. We find in particular that the more redundancy is present in parameter, the more intricately connected is its functional equivalence class.
We call for future work to seek out, catalogue, and thoroughly investigate sources of structural redundancy in more sophisticated neural network architectures; and to further investigate the role these parameters play in deep learning.
Acknowledgements
Contributions (1), (2), and (3) also appear in MFR's 's minor thesis Starugia-Roberte 2022, $5). MFR was affiliated with the School of Computing and Information Systems the University of Melbourne while completing this research, but is currently affiliated with the University of Cambridge.
We thank Daniel Murfet for helpful feedback during this research and this manuscript.
MFR received financial support from the Melbourne School of Engineering Foundation Scholarship and the Long-Term Future Fund while completing this research.
10

---

References
Francesca Albertini and Eduardo Sontag. For neural networks, function determines form. Tech- nical Report SYCON-92-03, Rutgers Center for Systems and Control, 1992. Expanded version of Albertini and Sontag (1993a). Cited on page
Francesca Albertini and Eduardo D. Sontag. For neural networks, function determines form. Neural Networks, 6(7):975-999) 1993a. Access via Crossref. Cited on pages 2 and 11.
Francesca Albertini and Eduardo D. Sontag. Identifiability of discrete-time neural networks. In Proceedings of the European Control Conference 1993, volume pages 460-465. European Control Association, 1993b. Access via Francesca Albertini. Cited on page 2.
Francesca Albertini and Eduardo D. Sontag. Uniqueness of weights for recurrent nets. In Systems and Networks: Mathematical Theory and Applications: Proceedings of the International Sympo- sium MTNS 1993, volume II, pages 599-602. Akademie Verlag, 1993c. Access via Francesca Albertini via Eduardo D. Sontag. See also extended version, access via Eduardo D. Sontag. Cited on page 2.
Francesca Albertini, Eduardo D. Sontag, and Vincent Maillot. Uniqueness of weights for neural networks. In Artificial Neural Networks for Speech and Vision, pages 1 113-125. Chapman & Hall, London, 1993. Proceedings of a workshop held at Rutgers University in 1992. Access via Eduardo D. Sontag. Cited on pages and 2.
Joachim Bona- Pelliseeli,,,,r, François Bachoc, and François Malgouyres. Parameter identifiability of deep feedforward ReLU neural network. 2021. Preprint 1XXv:22122112822 [math.ST]. Cited on page 2.
Johanni Brea, Berfin Şimşek, Bernd Illing, and Wulfram Gerstner. Weight-space symmetry in deep networks gives rise to permutation saddles, connected by equal-loss valleys across the loss land- scape. 2019, Preprint arXiv:1907.022 [es.LG]. Cited on page 9.
Stephen Casper, Xavier Boix, Vanessa D' Amario, Ling Guo, Martin Schrimpf, Kasper Vinken, and Gabriel Kreiman. Frivolous units: Wider networks are not really that wide. In Proceedings of the Thirty-Fifth AAAI Conference on Artificial Intelligence, volume 8, pages 6921-6929, AAAI Press, 2021. Access via Crossref. Cited on page 10.
An Mei Chen and Robert Hecht-Nielsen. On the geometry of feedforward neural network weight spaces. In Second International Conference on Artificial Neural Networks, pages 1-4. IET, 1991. Access via IEEE Xplore. Cited on page 2.
An Mei Chen, Haw-minn Lu, and Robert Hecht-Nielsen. On the geometry of feedforward neural network error surfaces. Neural Computation, 5(6)9110-927, 1993. Access via Crossref. Cited on page 2.
Matthew Farrugia-Robent Structural Degeneracy in Neural Networks. Master's thesis, School of Computing and Information Systems, The University of Melbourne, 2022. Access via Matthew Farrugia-Roberte Cited on page 10.
Matthew Harugia-Robbrte Computational complexity of determining proximity to compressible neural networks. 2023. Preprint wrXiv:2306.02833 [es.LG]. Cited on pages 1, 9, and 10.
Charles Fefferman. Reconstructing neural from its output. Revista Matemática Iberoameri- cana, 10(3):507-555, 1994, Access via Crossref. Cited on page 2.
Charles Fefferman and Scott Markel. Recovering feed-forward net from its output. In Advances in Neural Information Processing Systems 6, pages 335-342. Morgan Kaufmann, 1993. Access via NeurIPS. Cited on page 2.
Kenji Fukumizu. regularity condition of the information matrix of a multilayer perceptron net- work. Neural Networks, ((5):871-777 1996. Access via Crossref. Cited on pages 1, 10, and 13.
Kenji Fukumizu and Shun-ichi Amari. Local minima and plateaus in hierarchical structures of multilayer perceptrons. Neural Networks, 13(3)317-327,, 2000. Access via Crossref. Cited on pages 2, 5, and 9.

---

Kenji Fukumizu, Shoichiro Yamaguchi, Yoh-ichi Mototake, and Mirai Tanaka. Semi- flat minima and saddle points by embedding neural networks overparameterization. In Advances Neural Information Processing Systems 32, pages 13868-13877 Curran Associates, 2019. Access via NeurlPS, Cited on pages 2, 5, and 9.
Timur Garipov, Pavel Izmailov, Dmitrii Podoprikhin. Dmitry P. Vetrov, and Andrew G. Wilson. Loss surfaces, mode connectivity, and fast ensembling of DNNs. In Advances in Neural Information Processing Systems 31, pages 8789-8798 8798. Curran Associates, 2018. Access via NeurIPS. Cited page 9.
Egbert Harzheim. Ordered Sets. Springer, 2005. Access via Crossref. Cited on page 13.
Robert Hecht-Nielsen. On the algebraic structure of feedforward network weight spaces. In Ad- vanced Neural Computers, pages 129-135. North-Holland, Amsterdam, 1990. Access via Cross- ref. Cited on pages 1 and 2.
Torsten Hoefler, Dan Alistarh, Tal Ben-Nun, Nikoli Dryden, and Alexandra Peste. Sparsity in deep learning: Pruning and growth for efficient inference and training in neural networks. Journal of Machine Learning Research, 22(241):1-124, 2021. Access via JMLR. Cited on page 10.
Vera Kürková and Paul C. Kainen. Functionally equivalent feedforward neural networks. Neural Computaaiion 6(3):543-555) 1994. Access via Crossref. Cited pages 1 and 2.
Rohith Kuditipudi, Xiang Wang. Holden Lee, Yi Zhang, Zhiyuan Li, Wei Hu, Rong Ge, and Sanjeev Arora. Explaining landscape connectivity of low-cost solutions for multilayer nets. In Advances in Neural Information Processing Systems 32, pages 14601-14610. Curran Associates, 2019, Access via NeurIPS. Cited on page 9.
Edmund Lau, Daniel Murfet, and Susan Wei. Quantifying degeneracy in singular models via the learning coefficient. 2023. Preprint arXiv:2308. 12108 [sta Cited on page 10.
Mary Phuong and Christoph H. Lampert. Functional vs. parametric equivalence ReLU networks. In 8th International Conference Learning Representations. OpenReview, 2020. Access via OpenReview. Cited on pages and 2.
Stefan M. Rüger and Arnfried Ossen. The metric structure of weight space. Neural Processing Letters, 5(2):1-9, 1997. Access via Crossref. Cited on pages and 4.
Berfin Şimşek. François Ged, Arthur Jacot, Francesco Spadaro, Clément Hongler, Wulfram Gerst- ner, and Johanni Brea. Geometry of the loss landscape in overparameterized neural networks: Symmetries and invariances. Proceedings of the 38th International Conference on Machine Learning, pages
PMLR, 2021. Access via PMLR. Cited on pages 2, 5, 6, and 9.
Pierre Stock and Rémi Gribonval. An embedding of ReLU networks and an analysis of their identi- fiability, Constructive Approximation, 2022. Access via Crossref. Cited on page 2.
Héctor J. Sussmann. Uniqueness of the weights for minimal feedforward nets with given input- output map. Neural Networks, 5(4):589-593, 1992. Access via Crossref. Cited on pages 1, 2, 3. 4, 5, 13, 14, 15, and 16.
Vemer Vlačié and Helmut Bölcskei. Affine symmetries and neural network identifiability. Advances in Mathematics, 376: 107485, 2021. Access via Crossref. Cited pages 1 and 2.
Verner Vlačié and Helmut Böleskei. Neural network identifiability for family of sigmoidal non- linearities. Constructive Approximation, 55(1):173-225 2022. Access via Crossref. Cited on page
Sumio Watanabe. Algebraic Geometry and Statistical Learning Theory. Cambridge University Press, 2009. Cited on pages and 10.
Susan Wei, Daniel Murfet, Mingming Gong, Hui Li, Jesse Gell-Redman, and Thomas Quella. Deep learning is singular, and that's 's good. IEEE Transactions on Neural Networks and Learning Sys- tems, 2022. Access via Crossref. To appear in upcoming volume. Cited pages 1 and 10.
12

---

A Generalising to multi-dimeensionll inputs and outputs
In this appendix, we consider a slightly more general family of architectures than that introduced in Section 3. Namely, we consider a family of fully-connected, feed-forward neural network archi- tectures with n E N+t input units, m E N+ biased linear output units, and single hidden layer of h N biased hidden units with the hyperbolic tangent nonlinearity. With minor modifications, de- scribed in the remainder of this appendix, all definitions, algorithms, theorems, and proofs directly generalise from the case n m
arbitrary n and m.
Multi-dimensional architecture. Let E N+, m E N+, and h E N. Define the generalised pa- rameter space W,," 1o++++)))mmmmm The weights and biases of the network's units are encoded in the parameter vector in the format (41, b,, C1,-..,,,, bb, Ch2 d) = w. E -W.".. where for each hidden unit i 1, ,..... h there is an outgoing weight vector a, E R an incoming weight vector by E IR", and a bias & E R; and d E In is an output unit bias vector containing one bias value for each output unit. This time, indexes a multi-dimensional mathematical function fu I R defined as follows:
h
[1((x) = d
Σ
Og Lanh(b; x + c.).
Note that we use the same tuple notation and ordering (a1, bj, Ci.... ah. by, Ch. d) but now the a, the 6;. and d all denote multi-com coutii-ommpoent vectors. Accordingly, in Equation (2), by and x are now multiplied using the inner (dot) product, rather than scalar multiplication, since they are both vectors in R... Moreover, a; E IR'' as a vector is to be multiplied by the scalar tanh(b; x G). That is, the sum over vectors of contributions to output units from each hidden unit.
To generalise the results of the main paper to this setting the first change necessary is to replace all mentions of scalar weights with these vectors of weights, and other similar changes such reading the literal zero as vector zero where appropriate.
Signing and sorting incoming weight vectors. The lexicographic order on RR,, denoted x, is a relation such that for u, U E R", 3 if and only if u = v or, in the first index i = 1, 1,..., where u and differ, uN < u.. From this definition we follow the usual conventions in defining <, >, and z. Finally, define the lexicographic sign of R", denoted signlex()) as follows:
+1
(0 0),
sigmle(()
0
(v 0),
-1
(v < 0).
The parameter canonicalisation algorithm and some of the other theorems and proofs make repeated use of the signs of incoming weight vectors. The lexicographic sign satisfies the requisite properties of the scalar sign function in these uses and so the second change necessary to generalising the results is to replace uses of sign(-) with uses of (.))
This lexicographic order relation is of course also a total order (see, e.g., Harzheim, 2005, Theorem 4.1.11). Therefore, it allows one sort list of vectors. Sorting units by decreasing incoming weights is a key step in Stage 3 of Algorithm 4.1, and so the third change necessary is to use decreasing lexicographic order (>) in this stage.
Generalising Sussmann's equivalence theorem. The proofs in the main paper rely on the re- sults of Sussmann (1992) on the equivalence between reducibility and non-minimality, and the fact that irreducible functionally equivalent parameters are related by unit negation and exchange trans- formations. Sussmann (1992) studied a setting with multiple input units but only single output unit. Lemmas A.I and A.2 generalise these results to the multi-output setting." The final necessary change to generalise the results in the main paper is to replace all references to Sussmann's 's results with references to Lemma A. or Lemma A.2.
'The proof's reduce the multi- output case to the single-output case, so they still rely on the results of Suss- mann (1992). A generalisation similar to Lemma A.I is given by Fukumizu (1996).
13

---

The definitions of unit negation and exchange transformations, reducibility, and non- monimiimal all generalise arbitrary n and m with the above-mentioned changes. These definitions are repeated here for convenience.
m for i 1,....;h. h, where unit negation transformation is function o; Wx. Wh."
d: 0:(01, b1, .4, by, Ch d) = (a1, b1, C1,..., Oi bs -C,.... "a....,a,.. ah. by. Cha d).
A unit exchange transformation is a function 7, Wh + Wh. for i.j = 1, h, where Te. b1, C1...........................C.. ..... Oh, bb, Cha d) (a1, by, C1 C....... C.
aj, bj. Cjy a1+1,
Cj-1. a; bi, Cs. aj+1. Oh, bb, Ch. d).
parameter w (a1, b C1, Oh, bn, Ch. d) W, reducible if and only if satisfies any of the following conditions (otherwise, w is irreducible):
(i) a; = 0 for some i,
(ii) d; for some i,
(iii) (bi, Cx) = (bj, for some i + j.
(iv) (b,,c,) = (-bg. -cj) for some i i j.
A parameter Wii is non-minimal if and only if w is functionally equivalent some w/ tr with fewer hidden units h' < h.
Lemma A.1. For Wh, reducible if and only if w is non-minimal.
Proof. (+): A smaller functionally equivalent parameter can be constructed as follows.
ii If 0 for some i, then hidden unit fails to contribute the function. Construct functionally equivalent parameter w/ E W with hidden unit i omitted:
w = (a1, cI........ a4- bi G- 41+1, hi+1, G+1l. by, Ch, d)-
(ii) If O for some then hidden unit i contributes only a constant to the function. Construct functionally equivalent parameter w' E W with hidden unit i omitted and the output unit bias vector changed to compensate:
w' C1,,,,,,.....a a; -, G- 1 be+1, b G-1...... Cn. bn. Ch. + a, tanh(c;)).
(iii) If (b, .) = (bj,cz) for some * j. then hidden units i and j contribute proportionately. They be combined into single unit (say, with the same incoming weights and bias, and a combined outgoing weight vector. Construct a functionally equivalent parameter w' h- accordingly:
w' = =aaa, 01,,11... ( a1t1. ..-.. + ai by, Cj. ait1....,,........... ahy bn Ch. d).
(iv) If (b,, c;) = - (byj, Gj) for some i ل j. then hidden units i and j contribute in negative proportion. Due to the odd property of tanh they can be combined into single unit (say, ,) with incoming weight and bias vectors (b,,c,) and combined outgoing weight vector. Construct new parameter w' W accordingly:
w (a1, C1 ... G- ai+1, ................................ G-1, a, bj Cj aj+1, ah, bn, Cha d).
In all cases, the new parameter w/ E Wh has fwr = fu. so w is non-minimal.
(=): We reduce to the single-output case and apply the result of Sussmann (1992) to show that w satisfies least of the reducibility conditions.
To reduce to the single-output case, we introduce some notation. From the function fw : R - R
define a series of component functions f fer
aim) R" - R such that for x E R",
fa)( (ra) (x), (x) ".., (m))
14

---

m
fw
fw
Figure 3: The connection graphs of the component functions of fa- Included units and weights are solid. The hidden units of each network share the same incoming weights (and biases, not shown).
Each of these component functions is simple neural network function in architecture with n input units and 1 output unit, corresponding to a subgraph of the connection graph the original neural network, illustrated in Figure 3.
Denote the corresponding (overlapping) subvectors of w E Wh"" w(1)). ((m) Wh That is, for J ,,.
W = (ai,j,; by, Cl..... Oh,ps byy Cha dn) E Wm..
Now, let w' (ai,, b1, c ....... dn b/ ,,' E WA such that fw = fw where h' is the smallest number of hidden units required to implement fw (h' < h by assumption non-minimality). Apply the same decomposition fw' to define f(!) ".... f(m., and to define w/ w(i)..... ................................ ((m) Wm.
Apply the results of Sussmann (1992) follows. Since fur fw', (( ((()) for p= ..... follows that for each W(()) W(() is a functionally equivalent parameter using fewer units. Therefore, the reducibility conditions (in the special case of m = 1) must hold for each W(n) (Sussmann, 1992).
Since conditions (ii-i) only depend incoming weights and biases, if any of these conditions hold for any (()) then they must also hold for w itself (which shares the same incoming weights and biases), and the proof is complete. It remains only to consider the case in which conditions (ii-iv) fail to hold for any W(x), and to show that condition (i) holds for w itself in this case.
IR" - R the function 1, 1,...,'' denote by 4/ R" R the function (x) = tanh(b'xx + <4). Then, since we have ruled out reducibility conditions (ii-iv) for no SP; constant (ii) and no two are proportional (iii, iv). The same holds for the condininno (i-
We must introduce yet further notation. For i 1,.... h denote by sp, (z) tanh + G.). Similarly for
iv) do not hold for w(n)> since /i' was assumed to be minimal. Yet, for = 1,..., m, the linear
combination of functions
W
dn + Σ ain d, - Σ dy j=1
p(u)
p
1=1
yields the zero function. This linear combination remains when excluding those terms with a; 0 or a 0. Applying the same reasoning as that Sussmann (1992). due the independence property of the hyperbolic tangent function (Sussmann, 1992, Lemma 3.1) the remaining terms must be bijection, such that
P: = +
for some j with a'S, * 0 for each with ot + O.
To complete the proof, note that these relationships (3) between the units of w and w' are indepen- dent of However, the relationships are "exclusive" in the sense that no two Pi can be proportional to the same vbj. else they would also be proportional each other (ruled out above). Since there are only J units v1,..., Phr, it follows that there must be one hidden unit (actually at least // many units) for which aip 0 for all = l.....9 m (allowing ; to avoid any such relationship). D
That is, a; (0i,1..... m.,m) O, satisfying condition (i) for as required.
15

---

Let w Wh be irreducible, and let w' Wh. If w and w' are functionally equivalent then there exists a compositional chain of unit negation and exchange transformations, collectively transformation T Wh - Wh such that w' = (ww).
Lemma A.2.
Proof. Once again, we reduce to the case m and appeal to Sussmann (1992).
Suppose w' E [[v]]. Introduce the same decomposition of the two neural networks as in the proof of Lemma A.I, namely, the component functions f ..... (um s fo...... s..., ((m)) implemented by the
parameter subvectors w(1):-.. W(m); (()) -. w(m (m) Wh (cf. Figure 3).
For μ = ,,..., m, since fw = fw', we have that f(( = plr). Now, ((x) and w/()) are not necessarily irreducible, but if they are reducible then it is only by condition (i), since W(e) and w(y)) have the incoming weights and biases of and w' respectively irreducible because, with the same number of units as w, is necessarily minimal, and irreducibility follows by Lemma A.I). Remove such units with zero outgoing weight from W(w) and w(y)) to
is irreducible by assumption; w' is
produce new, functionally equivalent irreducible parameters "(H): ((w) W ra) Now by Sussmann (1992, Theorem 2.1) there exists a chain unit negation and exchange transformations Tμ such that (()) T. (m))))
For each μ., Tμ implies a relationship between the units of w(r) and w/y with nonzero outgoing weights, including possible negations and permutations of these units. This same relationship must hold between those units of w and w' since they share incoming weights and biases with ((()) and w/,, and (since w is irreducible, conditions (ii-iv)) these incoming weights are nonzero and the incoming weight and bias vectors are absolutely distinct between units of the same parameter. Moreover, all units are involved some such relationship because no unit of w w' can have zero outgoing weight vector by reducibility condition (i).
So, one can construct from these implied relationships composition of unit negation and exchange transformations relating w and w' as required.
16


Reviews:
[{"review_id": "5RDNq0e5Qa", "reviewer": "Reviewer_virJ", "summary": "The paper addresses fully connected feed-forward neural networks (NNs) with a single hidden layer and the hyperbolic tangent activation function. A parameter is thus a vector of weights and biases. It considers reducible parameters, which means that a NN with strictly less neurons can implement the same function. The paper provides a \"canonicalise\" procedure taking a parameter as input and yielding a canonical parameter that implements the same function. Also, if two parameters implement the same function then the output of the procedure is the same for them.\n\nBased on this procedure, the paper characterizes the set of parameters that implement the same function as a given parameter. It shows that this set is piecewise linear path-connected, and if the set is defined with respect to a ``sparse'' parameter, then the number of linear pieces of the diameter of the set is bounded by 7.   ", "questions": "The content presented is overall clear.\nThe statements of the theoretical results seem clear.\nSome proofs are a bit technical and more difficult to follow. The authors could consider adding pedagogical content/details on the proofs in the appendix, if they see fit.", "limitations": "The authors adequately discuss the main limitation (a single hidden layer) in the discussion.\nI do not see potential negative societal impact.", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper is well written. The figures are very clear and help understanding.\n\nThe theoretical topic is well motivated and well connected to the literature. \n\nThe theoretical results are interesting, and so are the proofs in my opinion.", "weaknesses": "The main weakness is that the paper is restricted to a single hidden layer. \nAlso, the hyperbolic tangent activation function is studied, while ReLU would have been more relevant, in my opinion.\n\nFor these two weaknesses, it seems they cannot really be fixed in the frame of this paper, as considering multiple hidden layers and/or ReLU would probably change the nature of the results and the proofs.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank reviewer virJ for their thorough review and for their kind words about the clarity of the paper. We appreciate the reviewer's feedback that some of the more technical elements of the proofs were difficult to follow. We are interested in broadening the accessibility of the paper to the extent possible. If possible, we would appreciate if the reviewer noted any specific sections of the proofs that were particularly difficult to follow? This would allow us to efficiently allocate our resources towards clarifying the presentation where it is most needed. Nonetheless, we are exceedingly pleased to hear that the reviewer found our results and proofs interesting.\n\nThe main concern of the reviewer appears to be the limited relevance of the setting studied in the paper. We refer the reviewer to the top-level author rebuttal where we have outlined how we see the results as forming an important component of a more general study of redundancy in more modern architectures.\nIt's true that the results and proofs will have to change in future work in this direction. However, we contend that these changes will be more like 'expansions' than fundamental changes. New methods will of course be needed, but we believe our contributions will still be useful.\nWe will reiterate that with this paper we believe we have identified a self-contained sub-problem (fully analysing the simple architecture) which fits (with clear presentation) into a single conference paper; and that we feel addressing the additional complexity arising from other architectural extensions can be best achieved with separate submissions.\n\nWe hope that this defense of the limited scope of the paper might earn the reviewer's reconsideration of their 'borderline' recommendation to accept the paper. However, either way, once again, we thank the reviewer for their attention and consideration.\n\n\n"}, {"review_id": "8vbzwcPRXK", "reviewer": "Reviewer_NhDx", "summary": "This paper deals with functional equivalence problems in neural networks, i.e., the characterization\nof all neural networks that lead to the same given output function. This is a problem that has been\nstudied since the early 1990s, including work by the fields medalist Charles Fefferman. \nThe present paper considers single-layer networks with tanh-nonlinearity and puts forward\na completely new perspective by paying attention to reducible parameters. ", "questions": "it would be interesting to see the authors' thoughts on whether the algorithmic component of the paper can\nbe extended to multi-layer networks", "limitations": "none", "rating": 10, "confidence": 5, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "The paper tackles a problem that has been studied in various guises for over 3 decades, and\nputs forward a completely new vantage point, by considering reducible parameters. This leads\nto rich insights into the functional equivalence problem. In addition, the paper exhibits a strong algorithmic component,\nspecifically by providing an algorithmic characterization of redundancies and connecting the underlying\ntheory to the beautiful concept of piecewise-linear path connected sets.\n", "weaknesses": "could not find any", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank reviewer NhDx for their review of our work and for their generous praise of our contributions. We appreciate the reviewer's acknowledgement of our novel perspective on the functional equivalence problem in particular.\n\nWe do acknowledge that our work definitely has important limitations in scope (as discussed in the top-level author rebuttal, and as raised by the other reviewers and acknowledged in our rebuttals). Chief among these limitations is the study of the simple architecture---to the reviewer NhDx's question. We refer the reviewer to our top-level author rebuttal for our detailed thoughts on the path to extending these results to more general architectures.\n\nWe will add for the interest of reviewer NhDx that our intuitions in this direction are influenced by the work of Fefferman (studying the *irreducible* multi-layer tanh case) and also the recent work of Vlačić and Bölcskei (e.g., \"Affine Symmetries and Neural Network Identifiability\", *Advances in Mathematics* 376, DOI 10.1016/j.aim.2020.107485) generalising Sussmann's reducibility conditions substantially (to networks with arbitrary feed-forward connection graphs). We are excited to draw on these works to push beyond the assumption of irreducibility and continue to explore the rich equivalence structure of reducible networks in more general architectures in future work."}, {"review_id": "AHheuLmJVt", "reviewer": "Reviewer_HCLr", "summary": "This paper focuses on the study of functional equivalence classes in neural networks, specifically in fully-connected, single-layer networks with hyperbolic tangent activation. It  presents a Canonicalisation algorithm for reducible networks to determine canonical representative parameters for different functional equivalence classes. Finally, it explores the concept of connectivity between these classes.", "questions": "The practical contribution of the path connectivity analysis was furthermore unclear for me. What does this analysis mean for learning or architecture design? Is it an argument for sparse training or initialization to reach an architecture with a majority of blank units?\n\nI believe the reference format is incorrect; I believe that NeurIPS uses number-based references instead of name, year.", "limitations": "The paper does not adequately address the limitations induced by the narrow focus on a highly theoretical architecture. The sections on \"towards modern architectures\" and \"functional equivalence and deep learning\" offer insight into the potential application of the analysis to contemporary deep neural networks, but are not comprehensive.", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper gives a well-motivation exploration of functional equivalence classes in neural networks. It provides a well-written definition of these classes, introducing an analysis of connectivity between them. The article references foundational works on simple neural networks and incorporates recent literature in the field of deep learning, enhancing the credibility and relevance of the study.", "weaknesses": "The weaknesses of this article stem from the focus on a very theoretical network type with implausible parameters. While a useful theoretical case, further discussion on the applicability of this analysis to modern architectures would benefit the article.\n\nSpecifically, a drawback of the article is its narrow focus on a specific architecture type, namely single-layer networks with tanh activation, a single input and single output. While it does discuss expanding the input and output space, it fails to address how the findings extend to more complex multi-layer networks or different activation functions commonly used in modern architectures. By limiting the scope of investigation, the article has limited application to contemporary neural network design and learning.\n\nThe proposal of the Canonicalisation algorithm also presents limitations. The algorithm relies on exact equivalence between weights or to 0, which is rarely achievable in practice, especially for weights trained through backpropagation. This raises questions about the adaptability of the Canonicalisation algorithm to networks trained through backpropagation, and whether it can provide meaningful results. The article would greatly benefit from addressing these concerns and exploring potential adaptations for networks trained through backpropagation.\n\nIt appears to me that the Canonicalisation algorithm is a unit pruning / neuron removal algorithm. While Kuditipudi et al. is cited, the article would benefit from considering and discussing other neuron removal methods, specifically the following which focus on individual neurons in fully-connected layers:\n\n+ Mao Ye, Chengyue Gong, Lizhen Nie, Denny Zhou, Adam Klivans, and Qiang Liu. Good subnetworks provably exist: Pruning via greedy forward selection. In Proceedings of the 37th International Conference on Machine Learning, pp. 10820–10830. PMLR, 2020.\n+ Xiaocong Du, Zheng Li, Yufei Ma, and Yu Cao. Efficient network construction through structural plasticity. IEEE Journal on Emerging and Selected Topics in Circuits and Systems, 9(3):453–464, 2019.\n+ Lemeng Wu, Bo Liu, Peter Stone, and Qiang Liu. Firefly Neural Architecture Descent: a General Approach for Growing Neural Networks. In Advances in Neural Information Processing Systems, volume 33, 2020.\n\nAdditionally, GradMax is a method which adds new neurons, similarly to Fukumizu and Amari, which was cited. The new neurons do not impact the function of the network upon addition by setting fan-in weights to 0. This case doesn't seem covered by remarks 5.3-5.5.\nEvci, Utku, et al. \"GradMax: Growing Neural Networks using Gradient Information.\" International Conference on Learning Representations. 2021.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank Reviewer HCLr for taking the time to thoroughly review our work. We aim to respond here to the several weaknesses and questions raised. Our question for the reviewer is, if the discussion of limitations and practical considerations in the paper were expanded along these lines, would the reviewer reconsider their recommendation to reject the paper, in line with some of the other reviewers who have recommended varying degrees of acceptance while acknowledging similar limitations? Or, is the reviewer's assessment of our contribution that it is fundamentally insufficient?\n\n**Weakness 1 (limited architecture):** Please see the top-level author rebuttal. In summary, we believe that our results form a meaningful step towards answers for more practical architectures.\n\n**Weakness 2 (exact canonicalisation):** We acknowledge the limitation of the canonicalisation algorithm based on exact equivalence. We do think of the canonicalisation algorithm as primarily a theoretical tool (as a basis for the characterisation of the full functional equivalence class), as the reviewer has noted.\n\nWe agree that two neural networks trained to approximate functional equivalence via backpropagation are unlikely to be detected as equivalent by this algorithm for numerical reasons. However, we still believe the canonicalisation algorithm has the potential to be developed in a direction that makes it more applicable as a diagnostic tool for this purpose. We offer the following observations:\n\n* After applying canonicalisation to two such networks, if they really are almost functionally equivalent, then their canonicalisations are likely to be close in parameter space, which can be easily detected (e.g. simply computing L_infinity distance).\n* One 'corner case' where the above claim does not hold is when the networks are near to the boundary of the 'canonical region', the slice of parameter space into which the canonicalisation algorithm maps parameters. Then it is possible that by slightly perturbing a network so as to cross this boundary, and then canonicalising the network, the resulting network will appear far away near another boundary of the region. Such networks will be approximately equivalent but not enough to be detected\n* It is simple to compute the L_infinity distance from a parameter to this boundary, which corresponds to finding the nearest reducible parameter.\n* This leads to a reliable test of approximate functional equivalence for parameters that are closer to each other than they are to the boundary of the canonical region.\n\nIf the reviewer finds this sketch compelling, then if accepted, we could add a formal version of these observations as an appendix.\n\n**Weakness 3 (missing discussion of pruning, gradmax):** We thank the reviewer for the detailed reference suggestions, which are highly appreciated. We must admit we are not deeply familiar with the literature on pruning, having come to this problem from the perspective of the literature on functional equivalence.\n\nWe acknowledge the apparent similarities between canonicalisation (which involves exact neuron removal) and unit-based network pruning, and also gradmax. One difference we have observed, stemming from the practical motivations of the pruning literature, appears to be that pruning methods usually accept an approximation of functionality (as long as loss is mostly concerned) while we have studied a setting of maintaining exact functional equivalence.\n\nWe believe that important future work will bridge these settings.\n\n* In our case that means studying approximate functional equivalence. We have another paper under blind review that makes this connection more precise, discussing a similar algorithm to the canonicalisation algorithm under the framing of 'lossless network compressibility' (we also study approximate relaxations of this problem and their computational complexity).\n* On the other hand, we think our systematic study of functional equivalence can inform pruning literature. Most pruning methods we have seen are based on removing units with little unilateral impact on the function output/the loss. Our study shows that sometimes units can be 'merged', but not unilaterally removed. Is reviewer is aware of any pruning approaches that take advantage of such 'higher-order' opportunities for unit removal? The closest we have seen is: Casper et al., 2019, \"Frivolous Units: Wider Networks Are Not Really That Wide\", discussing merging units with correlated outputs (cf. our reducibility condition (iii)).\n\n**Question 1 (relevance of path connectivity to learning or architecture design):** The main lessons we hope to draw from this kind of analysis is for learning (rather than architecture design). We have summarised in the Discussion our main takeaways in terms of bulk structure of the parameter space, and therefore the loss landscape, of our analysis. The main insight is in the large number of equivalent parameters that are reachable through short, simple paths from any given reducible parameter (see also: discussion in the top-level author rebuttal about how exactly we expect these insights to generalise). The main relevance of these results draws on the fact that in the overparameterised learning setting, many interpolating solutions are reducible, so this rich structure is inherited by the set of zero-loss parameters.\n\nOne could explore training methods that encourage reducibility directly, however, we are more interested in understanding whether or not existing learning methods already cause reducibility to emerge. This is one of our main motivations for theorising these sets of parameters: Now that we have characterised these sets, we can (in future work) conduct experiments aiming to observe their role in learning.\n\n**Question 2 (reference format):** The 2023 formatting instructions say \"Citations may be author/year or numeric, as long as you maintain internal consistency.\" We will follow any updated advice on the matter.\n"}, {"review_id": "ktslBYYwFF", "reviewer": "Reviewer_QPQA", "summary": "The paper introduces an algorithm to calculate the canonical equivalent parameter for a tanh feed-forward network.\nFurther, the full equivalent class of each parameter is given by a union of subsets constructed out of (inverse-)reductions and permutations of the canonical parameter.\nFinally, the paper shows that each functional equivalence class is path-connected and that the diameter of a functional class with rank at most the half of the number of units in the hidden layer is 7.\n", "questions": "196: What is a branch?\n\n265: Should it be ``connect $w'$ to $u$\"?\n\nWhat are the practical applications of this algorithm?\nCould you apply it to a loss function example you mention in 354?\nIf the parameters are real values, wouldn't they (and their absolute values) all be different from each other with probability equal to 1?\n\nCould you say anything about the loss function at different parts of the functional class that belongs to zero loss? Is it steeper at certain points in parameter space than others?\n\nCould you clarify and justify the remark in 362? What is this approximation? \n\nCould you give an example of a network that has rank higher than half of $h$ that is part of a functional equivalence class with a diameter more than 7?\n\nWhy does it matter what the diameter is of a functional equivalence class?\n\nWhy are networks that are in a functional equivalence class of diameter at most 7 of particular importance? \n", "limitations": "They are adequately discussed.\n", "rating": 5, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "The canonicalisation algorithm is novel for feed-forward networks with a tanh transfer function. Furthermore, this is the first result that characterizes the diameter of some of the equivalence classes of these networks.\nThe paper fully proves each theorem and includes useful visualizations for the path structure of a functional equivalence class.\nThe paper is overall very clearly written, with only a few possible minor typos.\nThe results presented here could be of importance to loss minimization problems.", "weaknesses": "The scope of applications of the algorithm and the theorems seems to be rather narrow or at least is not properly motivated to be broad.\n\nThe paper has very limited evaluation of the proposed algorithm and further uses of the results.\n\nIt is for example unclear in which practical application Algorithm 4.1 could be used, because of the real valued parameters being usually all different in a trained network, see Questions.\n\n", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank reviewer QPQA for the thorough review of our work. We appreciate in particular the kind comments about the clarity, novelty, and potential importance of the paper.\n\nWe hope to be able to sway the reviewer's recommendation to reject the paper by defending the relevance of the results in answer to the reviewer's weaknesses/concerns and specific questions about the motivations. (We are also happy to respond to the reviewer's technical questions.)\n\n**Weaknesses:**\n\n* We refer the reviewer to the top-level author rebuttal on the relevance of the results for a simple architecture. In summary, we believe that the results in this paper are a meaningful component of a generalisation to more practical architectures, and future work can explore the *additional* redundancy structure that emerges as new architectural components are added.\n* We refer the reviewer to our response to **weakness 2** of **reviewer HCLr** for a discussion of the relevance of the canonicalisation algorithm. In summary, this algorithm has clear theoretical utility (for characterising sets of equivalent parameters), which is the main motivation, but also it already has some practical utility for measuring approximate equivalence despite numerical issues and this could be developed further in future work.\n* Regarding the claim in the third question about real weights being unequal probability 1 (which is about the existence/relevance of reducible parameters at all, not just about the canonicalisation algorithm), this depends on the sampling distribution. We view it as unsettled whether weights in trained networks are actually unequal, since they are determined based on data rather than uniformly at random. As a weaker claim, it may also be true that weights are unequal but the course of training is still influenced by the local structure of the parameter space which may involve reducible networks.\n\n    If the reviewer has in mind a direct empirical evaluation of the hypothesis that reducible parameters and their functional equivalence classes are at all relevant to modern deep learning, we agree that such evaluation is an important direction for future work. Motivated by indirect evidence (such as empirical phenomena of compressibility in learned neural networks), we view our contribution as laying the groundwork for a principled study of reducibility and its role in practical deep learning.\n\n**Questions:**\n\n1. By 'branch' we simply mean 'if' or 'else if' in the algorithm. So the 'second branch' spans lines 8 to 11 of the algorithm (lines 144--147 of the manuscript). We apologise for being unclear. This appears to be standard programming terminology and we are not sure how to clarify it---we welcome specific suggestions.\n\n2. Yes. Thank you for pointing out this mistake and please accept our apologies for the confusion.\n\n3. We agree that with floating point parameters the canonicalisation algorithm will not usually face exact equality of any parameters. We refer the reviewer to the second dot point in response to weaknesses above. In summary, the algorithm has theoretical utility and potentially still has practical utility.\n\n4. It does appear that the steepness of the loss function varies for different equivalent parameters. As a simple example, in a one-unit network *a.tanh(bx)*, if *b* is perturbed around zero for large *a*, the function will vary much faster than if *b* is perturbed around zero for small *a*.\n\n5. If a parameter *w* achieves loss *L(w)* and is very close in parameter space to some reducible parameter *u*, then *L(u) ~= L(w)* by smoothness of *L*. Then there are paths of parameters with loss equal to *L(u)* throughout the whole functional equivalence class of *u*. In aggregate this implies a path of parameters from *w* to many other parameters with approximately equal loss to *L(w)*. If the reviewer has further questions we will be happy to elaborate.\n\n6. An example of such a high-diameter set is as follows. Consider a parameter with 10 units but rank 9, meaning there is exactly one 'spare' unit. There is a two-segment (at most) path from this parameter to a 'reduced form' of the parameter where a single unit is blank. Now to reach other equivalent parameters that are separated from this parameter by a permutation, it appears that the permutation must be implemented 'one swap at a time'. Some permutations can be implemented in a single swap. However, other permutations require some large minimum number of serial transpositions to be carried out in order to get all the units into the right place. Implementing one of these permutations to get to an equivalent reduced-form parameter will in general require potentially many more than 7 segments.\n\n    Note: We haven't formally ruled out shorter paths existing than those we construct, which only give upper bounds on shortest path lengths. However, we conjecture that shorter paths do not exist in this architecture based on our experience varying parameters, and we think we could probably prove lower bounds if we invested effort into this. It's not something we have considered yet.\n\n7. (and 8.) As discussed in the top-level author rebuttal, we do not actually think that the diameter itself or the number 7 are crucially important. Rather, by proving that the diameter is some small constant, we establish that all equivalent parameters are reachable with a small number of path segments. This suggests an extremely tightly connected network of equivalent parameters for these very-reducible parameters. Put another way: what matters is not the low maximum shortest path length but rather than number of short paths.\n\n    In this interest we are attempting to respond to empirical literature observing piecewise linear paths of low loss connecting learned networks (so-called 'mode connectivity' literature). We think our theoretical findings, by identifying one source of such paths, implicate (highly-)reducible networks in these phenomena. See also our Discussion section."}, {"review_id": "ED5m5w59Kn", "reviewer": "Reviewer_Wcpf", "summary": "The paper investigates the functional equivalence class for reducible neural network parameters and its connectivity properties. The authors focus on single-hidden-layer hyperbolic tangent networks, but the findings can be generalized to other feed-forward network components.", "questions": "In the paper, the authors focus on single-hidden-layer hyperbolic tangent networks. How do the authors think the results would change ifwe considered networks with multiple hidden layers or different activation functions?\n\nThe paper discusses the trade-offs between shortest path length and rank for different unit permutations. Could the authors provide more insights into how these trade-offs could impact the performance of neural networks in practical applications?", "limitations": "NA", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper provides a comprehensive understanding of the functional equivalence class for reducible neural network parameters. This can be key to understanding the structure of the parameter space and the loss landscape on which deep learning takes place.\n\nThe authors describe a complex union of manifolds, displaying rich qualitative structure. This includes a central discrete array of reduced-form parameters, connected by a network of piecewise linear paths, and various manifolds branching away from this central network.\n\nThe paper establishes that with a majority of blank units, the diameter of this parameter network becomes a small constant number of linear segments. This can be beneficial in understanding the trade-offs between shortest path length and rank for different unit permutations.\n\nThe paper also discusses the relevance of their findings to modern architectures and deep learning, suggesting that understanding reducible functional equivalence classes may be key to understanding these topics.\n", "weaknesses": "The paper is highly theoretical and may not provide immediate practical applications for those working with neural networks. The exact relevance of reducible parameters to these topics remains to be clarified.\n\n\nThe paper focuses on single-hidden-layer hyperbolic tangent networks, which may limit the applicability of the findings to more complex architectures.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank reviewer Wcpf for their thorough review of our submission. We think the strengths section provides a fair summary of the main contributions of the work and that the weaknesses section provides a fair summary of the main scope limitations of the work. We are pleased to reply to the reviewer's questions as follows:\n\n1. **Beyond single-hidden-layer tanh:** We refer the reviewer to the top-level author rebuttal where we outline a roadmap for generalising our algorithms and analysis beyond the this simple setting. In summary, we think that the results we have presented will form part of the picture in the more general case, but there is additional structure to redundant parameterisations in multiple layers and in other architectures (such as transformer architectures) that is yet to be explored.\n\n2. **Trade-offs between rank, permutation, and path length and their impacts on learning:** We appreciate this question as the topic seems interesting to us.\n    * We note that the impact on learning of the existence of short paths is through its implications for the structure of the loss landscape. Of interest in particular is the 'local' structure, in this case the number and arrangement of equivalent parameters a small number of path segments away from the current parameter. (See also the discussion in the top-level author rebuttal and the response to some other reviewers).\n    * We have identified two extreme points along this tradeoff:\n        * When the rank of the parameter is one less than the number of units, there are a small number of parameters nearby accessible by a small number of liner segments distance.\n        * When the rank of the parameter is half of the number of units, all equivalent parameters are within a small number of linear segments.\n    * The fundamental driver for these path lengths is the number of 'swap' manoeuvres required to implement the permutation separating the reduced form of two parameters. Some permutations require a large number of serial transpositions to implement, while all permutations can be implemented with a small serial depth if we can perform enough parallel transpositions in each serial step (this parallel transposition approach is at the heart of the diameter result).\n    * The ability to execute transpositions in parallel relies on the number of blank units that can be used as temporary registers for the weigh swap manoeuvres.\n    * Putting it together, this suggests that, the more blank units we have in the reduced form (the lower the rank or the original parameter), the more permutations we will be able to implement with a low equivalent parameters we will be able to implement with a small serial depth (due to more parallelisation), and therefore the more equivalent parameters will be reachable with a small number of linear path segments.\n"}]

Author rebuttal:
We thank the reviewers for their time spent thoroughly reviewing our submission. We have responded individually, and we wanted to take the opportunity here in the author rebuttal to expand on the discussion of limitations in the paper.

**Limitations of architecture generality:** All reviewers raised questions or concerns about the limitations in the scope of the study. As the reviewers have acknowledged, we discuss these limitations in the paper, but we are happy to expand here, and if accepted, to expand this discussion in the paper itself.

The main concern appears to be that the results for simple architecture (single-hidden-layer hyperbolic tangent networks) are not relevant to modern practice (with much more complex architectures), and it is unclear how our results will inform a study of the more practical setting.

Concretely, we are interested in generalising the main results (canonicalisation algorithm, characterisation of functional equivalence class, construction of piecewise linear paths) to architectures with features such as ReLU activation, multiple feed-forward layers, residual connections, or even other arrangements of neurons such as attention heads / transformer blocks.

Our claims are as follows:

1. A 'framing'-level contribution of our paper is a readily-generalisable set of questions for studying canonicalisation and functional equivalence beyond irreducible parameters. This appears to be a novel contribution to the theoretical literature.

2. **Our technical results on canonicalisation and functional equivalence are indeed meaningfully informative for future work** studying analogous questions in more complex architectures, as follows:
    * When it comes to functional equivalence in a single hidden layer, while our results have studied the tanh case specifically, most of the structure in the results is not specific to tanh. In particular, reducibility conditions (i) through (iii) are generic to feed-forward layers with any activation function, and only (iv) makes special use of the tanh 'odd' property. Other activation functions, such as ReLU, have their own additional symmetries giving rise to analogous conditions, such as positive linear scaling symmetry for ReLU. However, the methods used in algorithms and proofs for the ReLU case will be similar in how they account for reducibility conditions (i), (ii), and (iii). Therefore parts of our algorithms and proofs will generalise.
    * When it comes to studying the multi-layer case, we have made some headway by generalising our results to the single layer case with multiple inputs and outputs, as a multi-layer feed-forward network can be viewed as a composition of such functions. For this reason, part of canonicalisation in multi-layer architectures will involve canonicalising each layer independently, and then one must also account for interactions between layers. The per-layer canonicalisation will involve a straightforward application of our single-layer results, to which further algorithmic and theoretical analysis could be applied to resolve inter-layer redundancy. In other words, our results will form a significant part of the picture for the multi-layer case.
    * Even transformer blocks are composed of 'pieces' that look like feed-forward network layers, and so our results may partially inform generalisations in this direction.

3. When it comes to considering extensions of our path connectivity results, we expect our results to generalise, but not in the strong sense of 'the same connectivity properties will hold for other architectures'. We should clarify this point (here and in the paper):
    * We do *not* have any specific reason to expect that generalisations of the global connectivity properties or O(1) diameter result in other architectures. When additional sources of redundancy are introduced by extending the architecture, they may expand the functional equivalence class in areas that are not connected by piecewise linear paths.
    * However, while these 'global' connectivity results are the framing we use for technical statements, we believe the interesting implications of these results are *the fact that from a given (highly) reducible parameter, there are many equivalent parameters reachable with (a small number of) linear path components.* This 'local' perspective should not be disrupted by the additional of further equivalent parameters to the set, even if they are globally disconnected.
    * Along these lines we can already conjecture that modern architectures have rich 'local' connectivity properties in the functional equivalence classes of their reducible parameters, simply due to the presence of the same kinds of redundancy between neighbouring units within some layer of the architecture.

**Empirical relevance of reducible parameters:** We are grateful for our reviewers for following our motivation for the study of reducible parameters at all. Our reviewers do not appear to have taken major issue with the main gap in our motivation (for the most part), which is that so far we lack direct evidence of reducible parameters being encountered or approached during training in practical settings, which would be the basis for their relevance to modern deep learning theory. Our personal research agenda involves developing experiments to directly test our hypothesis that these parameters are relevant for learning. We consider this a priority over, say, investing more theoretical effort to systematically study new sources of redundancy arising from richer architectures.

---

Overall, we view hyperbolic tangent networks as a 'toy' architecture in which we have given a complete answer for *one part* of the general topic of redundancy in neural networks, and this is a concrete-enough architecture (historically sufficient) that we might also hope to be able to probe the relevance of reducible networks in practice.

Once again we thank our reviewers for their attention and their consideration.

Meta-review: By accounting for various redundancies in the parameterization of shallow neural networks, tanh activation, all different parameterizations implementing a given function are characterized and are further shown to be piecewise-linear connected.

At the beginning of the review process, opinions were somewhat divided whether the conceptual content, i.e., a framework offering general principles for functional equivalence under *reducible* parameters, outweighs the arguably oversimplified application, i.e., network architectures having single hidden layer and one input. After considerable discussions with the authors clarifying certain concerns, the reviewers have found the former to be a solid contribution to the topic of functional equivalence, mentioning further possible connections to existing work, and eventually unanimously voted for acceptance. The authors are encouraged to incorporate the important feedback given by the knowledgeable reviewers.
Decision: Accept (poster)